The Maximal Rank Conjecture

Size: px
Start display at page:

Download "The Maximal Rank Conjecture"

Transcription

1 The Maximal Ran Conjecture Eric Larson Abstract Let C be a general curve of genus g, embedded in P r via a general linear series of degree d. In this paper, we prove the Maximal Ran Conjecture, which determines the Hilbert function of C P r. 1 Introduction A central object of study in algebraic geometry in the past couple of centuries has been algebraic curves in complex projective space. (In this paper, we wor exclusively over C.) These can be described in two basic ways: Parametric Coordinates: We tae an abstract curve C of genus g, pic a line bundle L on C of degree d, and r + 1 linearly independent sections of H 0 (L) whose span is basepoint free. This gives rise to a degree d map C P r. Cartesian Coordinates: We tae a saturated homogeneous ideal I C[x 0, x 1,..., x r ] of height r 1. This gives rise to a curve V (I) P r. A natural question is: How do these points of view relate to each other? The Brill Noether theorem, proven by Griffiths and Harris [6], Gieseer [5], Kleiman and Lasov [11], and others, describes the space of parametric curves with general source: If C is a general curve of genus g, it states that there exists a nondegenerate degree d map C P r if and only if the Brill Noether number ρ(d, g, r) is nonnegative, where ρ(d, g, r) := (r + 1)d rg r(r + 1); and moreover, in this case, there exists a unique component of the Kontsevich space of stable maps M g (P r, d) that both dominates the moduli space of curves M g, and whose general member is nondegenerate. We term stable maps corresponding to points in this component Brill Noether curves (BN-curves). For r 3, it is nown that a general BN-curve is an embedding of a smooth curve, and so we may identify it with its image. There is a unique component of the Hilbert scheme containing the images of general BN-curves of degree d and genus g; by abuse of notation, we also term curves C P r corresponding to points in this component of the Hilbert scheme BN-curves. A specific instance of the above natural question is: What does the homogeneous ideal ( Cartesian equations ) of a general BN-curve loo lie, in terms of d, g, and r? 1

2 As a first step, one might as for the dimension of the graded pieces of the ideal, i.e. for space of polynomials of each degree that vanish on C, which can be described as the ernel of the restriction map H 0 (O P r()) H 0 (O C ()). The dimensions of these spaces are nown: ( ) r + dim H 0 (O P r()) = and dim H 0 (O C ()) = d + 1 g. The natural conjecture made originally by Severi in 1915 [7] when the Brill-Noether theorem was still a conjecture is: Conjecture 1.1 (Maximal Ran Conjecture). If C P r is a general BN-curve (r 3), the restriction maps H 0 (O P r()) H 0 (O C ()) are of maximal ran (i.e. either injective or surjective). Or equivalently, the dimension of the space of polynomials of degree which vanish on C is max ( 0, ( ) ) r+ (d + 1 g). Previously, many special cases of the maximal ran conjecture have been studied in the literature including the case of rational curves in P 3 by Hirschowitz [8]; the cases of nonspecial curves (i.e. the case d g + r) [4] and space curves [3] by Ballico and Ellia; the case of quadrics (i.e. for = 2) independently by Ballico [2], and by Jensen and Payne [9]; and many others. However, until now, a proof of the full conjecture was elusive. In this paper, we give the first proof in full generality: Theorem 1.2. Conjecture 1.1 (the Maximal Ran Conjecture) holds. More generally, we say a subscheme T P r satisfies maximal ran for polynomials of degree if the restriction map H 0 (O P r()) H 0 (O T ()) is of maximal ran. Since H 1 (O P r()) = 0, the long exact sequence in cohomolgy attached to the short exact sequence of sheaves 0 I C () O P r() O C () 0 implies that T P r satisfies maximal ran for polynomials of degree if and only if H 0 (I C ()) = 0 or H 1 (I C ()) = 0; the vanishing of H 0 (I C ()) being equivalent to the injectivity of the restriction map, and the vanishing of H 1 (I C ()) being equivalent to the surjectivity. In particular, the condition of satisfying maximal ran is open, and can therefore be approached via degeneration. Most cases of the Maximal Ran Conjecture that have proven thus far in the literature have used a specific degenerative approach due originally to Hirschowitz: 2

3 Degeneration to a reducible curve C C with C contained in a hyperplane H and C transverse to H. In this case, from the long exact sequence in cohomology attached to the short exact sequence of sheaves 0 I C P r( 1) I C C P r() I C (C H) H() 0, we conclude that to show H i (I C C Pr()) = 0 as desired, it suffices to show H i (I C P r( 1)) = Hi (I C (C H) H()) = 0. One can thus try to argue by induction on r and, reducing the desired result for (r, ) inductively to (r 1, ) and (r, 1). However, there are several critical issues with this approach, that have limited previous attempts to prove the maximal ran conjecture to special cases: 1. We need a uniform way to construct the reducible curves C C (previous methods were more or less ad-hoc). 2. We want: (a) Either H 0 (I C P r( 1)) = H0 (I C (C H) H()) = 0 H 1 (I C P r( 1)) = H1 (I C (C H) H()) = 0. (H 0 (I C P r( 1)) = H1 (I C (C H) H()) = 0, for example, proves nothing). (b) The fiber dimension of the map from the Hilbert scheme to M g at [C C ] to be ρ(d, g, r) + dim Aut P r, which forces C C to be a BN-curve. (c) The points C C to be general in H, so we may tae C and C each general. But each of these imply inequalities on the degree and genus of C, which do not in general have a solution. 3. This method relates maximal ran for C C to maximal ran for C and maximal ran for C (C H). Note that C (C H) is not a curve, so we need a stronger inductive hypothesis. But even worse, C and C must satisfy incidence conditions, so C and C H are not independently general and there is no nice description of C (C H) that doesn t reference the entire reducible curve C C. The ey innovations introduced some here and some elsewhere by the author to get around these difficulties are as follows: or 3

4 First Difficulty (uniformity): We leverage results on the interpolation problem for normal bundles, which determine the number of general points a BN-curve of given degree and genus can pass through (as well as variants which wor when some of the points are constrained to lie in a transverse hyperplane). If we find two curves C 1 and C 2 both passing through a finite set of points Γ, their union produces a reducible curve. Results on the interpolation problem thus let us build desired curves uniformly by showing certain systems of inequalities have integer solutions. The interpolation problem for normal bundles was studied in a sequence of papers by the author and others [1, 14, 16, 18, 19], which contains all the results on this topic that we shall need here. Second Difficulty (desiderata conflict): We relax our second desideratum, and show that certain reducible curves constructed as above are BN-curves, even when the fiber dimension of the map from the Hilbert scheme to M g is too large. This is done by first leveraging results on the interpolation problem for restricted tangent bundles to calculate this fiber dimension at certain other reducible curves of this form, and showing that for these other curves it is correct. Then we use iterative specialization and deformation to construct a broen arc in the Kontsevich space (which may not mae sense in the Hilbert scheme compactification) between the desired reducible curves and the other ones. Provided that the specializations are to smooth points of the Kontsevich space, this shows the desired curves are in the same component of the Kontsevich space, and thus the same component of the Hilbert scheme, as these other curves. This program is carried out in a sequence of papers by the author [12, 13, 15], which contains all the results on this topic that we shall need here. Third Difficulty (C (C H)): The ey idea to get around this difficulty is to degenerate to a 3-component curve C 1 C 2 C, with C a general BN-curve in a hyperplane H, and C 1 C 2 a BN-curve transverse to H such that [C 2 H] [C 1 C 2] Sym deg C 2 P r Sym #(C 1 C 2 ) H is general. We then smooth C 1 C 2 to a general BN-curve C, while preserving the incidence conditions with C. The method of Hirschowitz relates maximal ran for C C P r to maximal ran for C P r, which we can apply induction to; and to maximal ran for C (C H), which can be further specialized to C (C 1 H) (C 2 H). But now C 1 H is a set of general points, and C and C 2 H are independently general (and so can in particular be described without reference to the incidence conditions)! Since these hyperplane sections are independently general, they may be studied separately using results of the author in [17]. This construction is studied in Section 2 of the present paper. The upshot is that we can then argue by induction on the the following stronger hypothesis (note that taing n = ɛ = 0 recovers the maximal ran conjecture, so this proves Theorem 1.2 as desired): Theorem 1.3. Fix an inclusion P r P r+1, and let be a positive integer. Let C P r be a general BN-curve (which is alternatively permitted to be a general rational curve of degree d < r 4

5 if 3); D 1, D 2,..., D n P r+1 be independently general BN-curves which are required to be nonspecial if = 2 and r 4; and p 1, p 2,..., p ɛ P r be a general set of points, where { 2 if n > 0 and 5 and r 4; ɛ ɛ 0 := 0 otherwise. Then any subset of T := C ((D 1 D 2 D n ) P r ) {p 1, p 2,..., p ɛ } P r which contains C {p 1, p 2,..., p ɛ } satisfies maximal ran for polynomials of degree. As explained above, our argument will be by induction on r and ; we shall reduce Theorem 1.3 for (r, ) inductively to Theorem 1.3 for (r 1, ) and (r, 1). But first, for fixed (r, ), we may mae the following reductions: 1. Applying the uniform position principle and Lemma 2.5 of [17], it suffices to prove that T satisfies maximal ran in Theorem 1.3. (Theorem 1.3 is stated for any subset of T containing C {p 1, p 2,..., p ɛ } only because that is a more convenient inductive hypothesis.) 2. If n = 0 or r = 3 or = 2, then again by Lemma 2.5 of [17], we may reduce to the case ɛ = 0. (Note that we may always lower ɛ if desired, since general points impose independent conditions on any subspace of sections of O P r().) Notation: In the proof of Theorem 1.3, we write d and g for the degree and genus of C, and d i and g i for the degrees and genera of the D i. We also define h := ɛ ɛ 0 + n d i. i=1 Organization of the remainder of the paper: We begin, in Section 2, by studying the ey degenerations that we shall use in our inductive argument. Then in Sections 3 and 4, we establish Theorem 1.3 for space curves (r = 3) and quadrics ( = 2), leveraging results of Ballico and Ellia [3] and Ballico [2] which establish Theorem 1.2 in these cases; these will form the base cases for our larger inductive argument. In Sections 5 and 6 we study the easy cases of Theorem 1.3, where the restriction map is far from being an isomorphism (i.e. either the dimension of the source is much larger than the dimension of the target, or vice versa); here we leverage results of Ballico and Ellia which establish Theorem 1.2 for nonspecial curves [4]. Intuitively, these are the easier cases since the codimension in the space of all linear maps of those with non-maximal ran is highest when the dimensions are far apart. Finally, in Section 7, we use the machinery of the preceding sections to reduce Theorem 1.3 to a computation involving the existence of integers satisfying certain systems of inequalities. This computation is taen care of in Appendices A, B, and C. 5

6 Acnowledgements The author would lie to than Joe Harris for his guidance throughout this research, as well as Atanas Atanasov, Edoardo Ballico, Brian Osserman, Sam Payne, Ravi Vail, Isabel Vogt, David Yang, and other members of the Harvard and MIT mathematics departments, for helpful conversations or comments on this manuscript. The author would also lie to acnowledge the generous support both of the Fannie and John Hertz Foundation, and of the Department of Defense (NDSEG fellowship). 2 Degenerations In this section, we outline the main degenerations we shall use in the proof of Theorem 1.3. Definition 2.1. We say r,, d, g, d, g, h, h, and ɛ 0 satisfy I(r,, d, g, d, g, h, h, ɛ 0, ɛ 0) (respectively satisfy S(r,, d, g, d, g, h, h, ɛ 0, ɛ 0)) if ( ) ( 1)d + 1 g + h + ɛ r + 0 (respectively ) r + (1) ( ) d g ( 1)d + g + h h + ɛ 0 ɛ r r + 0 (respectively ) r +. (2) Proposition 2.2. Suppose that Theorem 1.3 holds for (r 1, ) and (r, 1), and let H P r be a hyperplane. Assume there exists: 1. A specialization of C to an interior BN-curve C = C C, with C contained in H and C transverse to H, with C C general, and with C of degree d and genus g. 2. Specializations {p ɛ 0 +1, p ɛ 0 +2,..., p ɛ} of {p ɛ0 +1, p ɛ0 +2,..., p ɛ }, and D i of each D i, with #({p ɛ 0 +1, p ɛ 0 +2,..., p ɛ} H) + #(D i H) = h h. 3. A specialization {p 1, p 2,..., p ɛ 0 } of {p 1, p 2,..., p ɛ0 }, with #({p 1, p 2,..., p ɛ 0 } H) = ɛ 0 ɛ 0. Assume also that for some deformation C of C, A := ( C D 1 D 2 D n {p 1,..., p ɛ} ) H H and B := C (( D 1 D 2 D n {p 1,..., p ɛ} ) (P r H) ) P r satisfy the assumptions of Theorem 1.3 or are the union of subsets hyperplane sections of general BN-curves, which are nonspecial if = 2. If r,, d, g, d, g, h, h, ɛ 0, and ɛ 0 satisfy either then Theorem 1.3 holds for T. I(r,, d, g, d, g, h, h, ɛ 0, ɛ 0) or S(r,, d, g, d, g, h, h, ɛ 0, ɛ 0), 6

7 Proof. Since C C is general in H, and C is transverse to H, we may arrange for C to pass through C C, so that C C is a deformation of C C. Since C C is an interior BN-curve by assumption, C C is also a specialization of C. Write T = C C ((D 1 D 2 D n) P r ) {p 1, p 2,..., p ɛ}. Then we have an exact sequence 0 I B ( 1) I T () I T H() 0. In particular, to show H i (I T ()) = 0, it suffices to show H i (I B ( 1)) = H i (I T H()) = 0. Since T H can be specialized to A, it thus suffices to show H i (I B ( 1)) = H i (I A ()) = 0, or equivalently that the restriction maps H 0 (O H ()) H 0 (O A ()) and H 0 (O P r( 1)) H 0 (O B ( 1)) are either both injective or both surjective. Since A H and B P r satisfy the assumptions of Theorem 1.3 or Theorem 1.3 of [17], we now by our inductive hypothesis that each of these maps is either injective or surjective. It thus remains to show that either or dim H 0 (O B ()) dim H 0 (O P r( 1)) and dim H 0 (O A ()) dim H 0 (O H ()), dim H 0 (O B ()) dim H 0 (O P r( 1)) and dim H 0 (O A ()) dim H 0 (O H ()). But these are exactly the conditions I(r,, d, g, d, g, h, h, ɛ 0, ɛ 0) and S(r,, d, g, d, g, h, h, ɛ 0, ɛ 0) respectively. Proposition 2.3. Suppose that Theorem 1.3 holds for (r 1, ), and let H P r be a hyperplane. Assume there exists: 1. A specialization of C to a general BN-curve C H. 2. Specializations {p 1, p 2,..., p ɛ} of {p 1, p 2,..., p ɛ }, and D i of each D i, such that B := ( D 1 D 2 D n {p 1,..., p ɛ} ) (P r H) P r is a union of subsets of hyperplane sections ) ( of general BN-curves, which are nonspecial if = 3, and has cardinality = r+ 1 ). If in addition (r+ r+ 1 A := C ( (D 1 D 2 D n {p 1,..., p ɛ}) H ) H satisfies the assumptions of Theorem 1.3, then Theorem 1.3 holds for T. Proof. Write T = C ((D 1 D 2 D n) P r ) {p 1, p 2,..., p ɛ}. Then we have an exact sequence 0 I B ( 1) I T () I A () 0. In particular, to show H i (I T ()) = 0, it suffices to show H i (I B ( 1)) = H i (I A ()) = 0. By Theorem 1.3 of [17] and our second assumption, we have H 0 (I B ( 1)) = H 1 (I B ( 1)) = 0. It thus suffices to note that, by our inductive hypothesis for Theorem 1.3 either H 0 (I A ()) = 0 or H 1 (I A ()) = 0. 7

8 Definition 2.4. We say that ɛ 0, ɛ 0, and t satisfy E(ɛ 0, ɛ 0, t) if t ɛ 0 ɛ 0. Proposition 2.5. If t 0, there exists a specialization {p 1, p 2,..., p ɛ 0 } of {p 1, p 2,..., p ɛ0 }, with #({p 1, p 2,..., p ɛ 0 } H) = ɛ 0 ɛ 0, so that {p 1, p 2,..., p ɛ 0 } H H and {p 1, p 2,..., p ɛ 0 } P r H P r are sets of general points, the second of cardinality at least t, provided that ɛ 0, ɛ 0, and t satisfy E(ɛ 0, ɛ 0, t). Proof. This is clear since {p 1, p 2,..., p ɛ0 } are general points, so can be specialized to an arbitrary set of points. Definition 2.6. We say that r, h, and h satisfy A(r, h, h ) if 0 h h r + 1. Proposition 2.7. There exist specializations {p ɛ 0 +1, p ɛ 0 +2,..., p ɛ} of {p ɛ0 +1, p ɛ0 +2,..., p ɛ }, and Di of each D i, with so that are sets of general points, and #({p ɛ 0 +1, p ɛ 0 +2,..., p ɛ} H) + #(D i H) = h h, {p ɛ 0 +1, p ɛ 0 +2,..., p ɛ} H H and (D 1 D 2 D n {p ɛ 0 +1, p ɛ 0 +2,..., p ɛ}) (P r H) P r D 1 H, D 2 H,..., D n H H is a set of hyperplane sections of independantly general BN-curves, for some integer h satisfying A(r, h, h ). Proof. Since general points in P r can be specialized to general points in either H or P r, we reduce to the case ɛ = ɛ 0, which follows from combining Lemmas 5.3 and 6.1 of [17]. Definition 2.8. We say that r, h, and h satisfy J(r, h, h ) if h r + h h, r + 1 satisfy K(r, h, h ) if h 2r + 1 and 0 h h, 8

9 satisfy L(r, h, h ) if satisfy M(r, h, h ) if and satisfy N(r, h, h ) if h 3r + 2 and h = 2, h 3r + 2 and r + 2 h h h r + h h r r + 1 h 2r + 2 r 2,. Proposition 2.9. There exist specializations {p ɛ 0 +1, p ɛ 0 +2,..., p ɛ} of {p ɛ0 +1, p ɛ0 +2,..., p ɛ }, and Di of each D i, with so that are sets of general points, and #({p ɛ 0 +1, p ɛ 0 +2,..., p ɛ} H) + #(D i H) = h h, {p ɛ 0 +1, p ɛ 0 +2,..., p ɛ} H H and {p ɛ 0 +1, p ɛ 0 +2,..., p ɛ} P r H P r D 1 H, D 2 H,..., D n H H and D 1 P r H, D 2 P r H,..., D n P r H P r are sets of subsets of hyperplane sections of independantly general BN-curves, provided that r, h, and h satisfy J(r, h, h ). Moreover, such specializations exist so that the second of these sets can be specialized to a set of subsets of hyperplane sections of independantly general nonspecial BN-curves if r, h, and h satisfy one of K(r, h, h ), L(r, h, h ), M(r, h, h ), or N(r, h, h ). Proof. Since general points in P r can be specialized to general points in either H or P r, we reduce to the case ɛ = ɛ 0. By combining Lemmas 5.4 and 6.1 of [17], we see such a specialization exists if r, h, and h satisfy J(r, h, h ); moreover, we can tae the second of these sets to be a set of subsets of hyperplane sections of independantly general nonspecial BN-curves if N(r, h, h ) is satisfied. It thus remains to consider the case when one of K(r, h, h ), L(r, h, h ), or M(r, h, h ) is satisfied. Note that d i d i ρ(d i, g i, r + 1) = (r + 1) (g i + r + 1 d i + 1). In particular, D i can only be special if its degree is at least 2r + 2. If K(r, h, h ) is satisfied, we thus conclude all D i are nonspecial. Similarly, if L(r, h, h ) or M(r, h, h ) is satisfied, then since a BN-curve in P r+1 has degree at least r +1, we thus conclude that either all D i are nonspecial or n = 1 and g 1 = d 1 r. Since the the desired result when all D i are nonspecial follows from Corollary 4.2 of [17], it remains to consider the case when L(r, h, h ) or M(r, h, h ) is satisfied, n = 1, and g 1 = d 1 r. 9

10 If L(r, h, h ) is satisfied, the result now follows from Lemma 5.3 of [17]. It thus remains to consider the case when r, h, and h satisfy L(r, h, h ) and not N(r, h, h ), i.e. when r h 3r + 2 and h r + 1 h h. 2 By the uniform position principle, the points of D 1 P r are in linear general position. We may therefore apply an automorphism of P r so that exactly h h r 1 of these points lie in H. Since the automorphism group of P r acts transitively on sets of h h r 1 points in linear general position, D 1 H can be assumed to be general; in particular, it is a subset of the hyperplane section of a general rational normal curve. It remains to see D 1 P r H, which is a subset of a hyperplane section of a general special BN-curve, can be specialized to a subset of a hyperplane section of a general nonspecial BN-curve. For this, we apply Theorem 1.8 of [12] to degenerate D 1 P r to a stable map f : D Γ P 1 P r, with f D a general BN-curve of degree d 1 r 2 and genus d 1 r 2 r 1 (which is in particular nonspecial), and f P 1 of degree r 2, and #Γ = r Since h h r 2 by assumption, we can arrange for D 1 P r H f(d) H. Definition We say that r, d, g, d, and n satisfy X(r, d, g, d, n) if g n (3) r(d d ) (r 1)g + (r 1)n r (4) n 1 0 (5) d n 0 (6) r + 2 n 0 (7) 2n + d d g r 1 0. (8) Proposition Let r 4 and d, g, and d be integers which satisfy (r + 1)d rg r(r + 1) 0, and suppose there exists an integer n which satisfies X(r,, d, g, d, n). Then there exists a BNcurve C C P r of degree d and genus g, with C a general BN-curve in a hyperplane H, and C a general rational curve of degree d transverse to H. Proof. Let n be some such integer. Write g = g n + 1 and d = d d. Upon rearrangement, (3) gives g 0, and (4) gives ρ(d, g, r 1) 0. We may therefore let C H be a general BN-curve of degree d and genus g. Note that C passes through r + 2 general points in H, by Corollary 1.4 of [1] if C is nonsepcial or Theorem 1.2 of [16] if C is special. In particular, C passes through a set Γ of n 1 general points in H (note that n r + 2 by (7), and that n 1 by (5)). Since d n by (6), and the hyperplane section of a general rational curve of degree d is a general set of d points by Corollary 1.5 of [15], we may thus let C be a general rational curve of degree d passing through Γ. By construction, C C is of degree d + d = d and genus g + n 1 = g. It thus remains to see it is a BN-curve; by Theorem 1.9 of [12], it suffices to see n r + 2 and d + n g + r. The first of these inequalities is just (7), while the second becomes upon rearrangement (8). 10

11 Definition We say that r, d, g, d, g, n, and t satisfy Y (r, d, g, d, g, n, t) if g 0 (9) (r + 1)d rg r 2 r 0 (10) (2r 3)d (r 2) 2 g 2r 2 + 3r 9 0 (11) g g n (12) r(d d ) (r 1)(g g ) + (r 1)n r (13) n 1 0 (14) d n t 0 (15) r(d d ) (r 4)(g g ) 2n 2r (16) 2n + d + g d g r 2 0. (17) Proposition Let r 4 and t 0, and d, g, d, and g be integers which satisfy (r + 1)d rg r(r + 1) 0, and suppose there exists an integer n satisfying Y (r,, d, g, d, g, n, t). Then there exists a BNcurve C C P r of degree d and genus g, with C a general BN-curve in a hyperplane, and C a general BN-curve of degree d and genus g transverse to H such that C H is a set of d general points in H, at least t of which do not lie on C. Proof. Let n be the minimal such integer. Note that (11) and (15), together with our assumption that t 0, imply Additionally, (15) implies (2r 3)d (r 2) 2 (g d + n) 2r 2 + 3r 9 0. (11 ) d n 0. (15 ) Since the left-hand sides of (11) and (15) are nonincreasing in n, it follows that n is also the minimal integer satisfying the system of inequalities (11 ), (12), (13), (14), (15 ), (16), (17). Additionally, note that (11) also implies (2r 3)(d + 1) (r 2) 2 g 2r 2 + 3r 9 0. We conclude the desired curve exists by applying Theorem 1.1 and Remar 1.2 of [13]. Lemma Let X P r be a subscheme of codimension at least 2, and X be a set of r + 2 points which are general in some nondegenerate component of the smooth locus. Then for every integer m 1, there exists a BN-curve C of degree rm and genus (r + 1)(m 1) whose intersection with X is exactly the reduced scheme. Proof. We argue by induction on m. When m = 1, we first note that Aut P r acts transitively on sets of r + 2 points in linear general position. Applying an automorphism to a rational normal curve, we may thus find a rational normal curve C passing through. It is a classical fact (and also an immediate consequence of Theorem 1.3 of [1]) that N C O P 1(r + 2) (r 1), and so N C ( ) O (r 1) P has 1 11

12 vanishing cohomology and is generated by global sections. We may thus deform C to a curve passing through which avoids any (excess) intersection with any subvariety of codimension at least 2. For the inductive step, we let C 0 be a BN-curve of degree r(m 1) and genus (r + 1)(m 2) whose intersection with X is exactly the reduced scheme. We then let 0 be a set of r + 2 general points on C 0. Applying our inductive hypothesis again, we may find a rational normal curve C 1 whose intersection with X C 0 is exactly the reduced scheme. Taing C = C 0 C 1 completes the proof, as this is a BN-curve by Theorem 1.6 of [12]. Definition We say that r, d, g, d, g, n, m, and t satisfy Z(r, d, g, d, g, n, m, t) if (r + 1)d rg r 2 r 0 (18) g (r + 1)m 0 (19) (2r 3)(d rm) (r 2) 2 (g (r + 1)m) 2r 2 + 3r 9 0 (20) g g n (21) r(d d ) (r 1)(g g ) + (r 1)n r (22) n 1 0 (23) (d rm) n t 0 (24) r(d d ) (r 4)(g g ) 2n 2r (25) 2n + d + g d g r 2 0 (26) 2(d rm) (r 3)(g (r + 1)m 1) (r 1)(r + 2) 0 (27) m 0. (28) Proposition Let r 4 and t 0, and d, g, d, and g be integers which satisfy (r + 1)d rg r(r + 1) 0, and suppose there exist integers n and m satisfying Z(r, d, g, d, g, n, m, t). Then there exists an interior BN-curve C 1 C 2 C P r of degree d and genus g, with C a general BN-curve in a hyperplane, and C 1 C 2 a BN-curve of degree d and genus g transverse to H such that C 2 H is a set of general points in H, at least t of which do not lie on C 2, and C 1 is either a BN-curve which is general independent from C 2 H or C 1 =. Proof. By Proposition 2.13, there exists a BN-curve C 2 C P r of degree d rm and genus g (r + 1)m, with C a general BN-curve in a hyperplane H, and C 2 a general BN-curve of degree d rm and genus g (r + 1)m transverse to H such that C 2 H is a set of d rm general points in H, at least t of which do not lie on C. Write Γ = C 2 C. By (20), the bundle N C 2 ( 1) satisfies interpolation. In particular, using (27), we have H 1 (N C 2 ( 1)( )) = 0 where C 2 is a set of r + 2 general points on C 2. We have the exact sequences 0 N C 2 C C 2 ( Γ ) N C 2 C ( ) N C 2 C C 0 0 N C 2 ( 1)( ) N C 2 C C 2 ( Γ ) 0 0 N C /H N C 2 C C N H C (Γ) O C (1)(Γ) 0, 12

13 where the s denote punctual sheaves, which in particular have vanishing H 1. Since Lemma 3.2 of [12] gives H 1 (N C /H) = 0, we conclude H 1 (N C 2 C ( )) = 0 provided H1 (O C (1)(Γ)) = 0. But since C H is a general BN-curve of degree d d and genus g +1 g n, we have either H 0 (O C (1)) = r or H 1 (O C (1)) = 0. In the second case, H 1 (O C (1)(Γ)) = 0 is immediate. In the first case, this implies via (26) that dim H 1 (O C (1)) = r χ(o C (1)) = n 2 (2n + d + g d g r 2) n. Since twisting up by a general point drops the dimension of H 1 when that dimension is positive (the Serre dual of the familiar statement that twisting down by a general point drops the dimension of H 0 when that dimension is positive), we conclude H 1 (O C (1)(Γ)) = 0 and thus H 1 (N C 2 C ( )) = 0. If m = 0, we tae C 1 = ; as H 1 (N C 2 C ( )) = 0, we have H 1 (N C 1 C 2 C ) = H1 (N C 2 C ) = 0. In particular, [C 1 C 2 C = C 2 C ] is a smooth point of the Hilbert scheme, and thus an interior curve as desired. If m 1, we apply Lemma 2.14 to let C 1 be a BN-curve of degree rm and genus (r+1)(m 1) whose intersection with C 2 is exactly. We then deform C 1 to be general in some component of the space of BN-curves passing through. By Theorem 1.6 of [12], both C 1 C 2 and C 1 C 2 C are BN-curves. Moreover, using our assumption that H 1 (N C 2 C ( )) = 0, Lemmas 3.2, 3.3, and 3.4 of [12] imply C 1 C 2 C is an interior curve, as desired. In Appendix A, we give code in sage to chec or create algebraic expressions for all inequalities that appear in this section. 3 Space Curves In this section, we prove Theorem 1.3 for space curves (r = 3); this will serve as one of the base cases for our larger inductive argument. Our argument here will also be by induction, this time on n. Recall that, as mentioned in the introduction, it suffices to prove maximal ran for T, and we may tae ɛ = 0. Our base case will be n = 0, for which this is a result of Ballico and Ellia if C is a BN-curve [3], and a well-nown classical fact if C is a general rational curve of degree d < r (in this case the restriction map is always surjective). We therefore assume for our inductive argument that n 1. Write d i = deg D i and g i = genus D i, and suppose without loss of generality that d 1 d 2 d n ; write d and g for the degree and genus of C. By our inductive hypothesis, the subscheme T n 1 := C ((D 1 D 2 D n 1 ) P r ) P r satisfies maximal ran for polynomials of degrees and 1. Moreover, if Λ P 3 is a general plane, and 2, then Theorem 1.3 of [17] implies T n 1 Λ = C Λ Λ satisfies maximal ran for polynomials of degree. Since T n 1 is positive-dimensional, an application of Theorem 1.5 of [17] completes the proof unless 3 and (d n, g n ) {(8, 5), (9, 6), (10, 7)}, and dim H 0 (O P 3( 1)) > dim H 0 (O Tn 1 ( 1)) and dim H 0 (O Λ ()) < 8 + dim H 0 (O C Λ ()), 13

14 or equivalently, ( ) + 2 n 1 ( 1)d g d i (29) 3 i=1 ( ) d. (30) 2 Note also that in this case, D n P 3 is the general complete intersection of 11 d n quadrics (c.f. Theorem 1.6 of [14]); in particular, we may specialize it to the union of the general complete intersection of 3 quadrics plus d n 8 additional general points. We may thus reduce to the case d n = 8. Since 3, the inequality (30) implies d ( ) ( ) 2 7 = 3; in particular, C cannot be a degenerate rational curve. We thus have ρ(d, g, 3) 0, or upon rearrangement: g 4d 12. (31) 3 Combining (31) with our assumption that d i d n 8 for all i, condition (29) implies ( ) d + 8n Rearranging and combining this with (30), we obtain In particular, if n 2, then d n. (32) , 6 14 which does not hold for any 3; consequently, n = 1. In this case, , 6 14 which does not hold for any 5; consequently, {3, 4}. For each, equation (32) gives upper and lower bounds on d; for each such d, equations (31) and (29) then give upper and lower bounds respectively on g. Using these bounds, it thus remains only to consider the case where n = 1, and D 1 = D n is a canonical curve; and either = 3 and or = 4 and (d, g) {(3, 0), (4, 0), (4, 1), (5, 2), (6, 4)}, (d, g) = (8, 6). If = 3 and (d, g) {(4, 0), (5, 2), (6, 4)}, then using our inductive hypothesis that C satisfies maximal ran for cubics, we see that C lies on at most a 7 dimensional family of 14

15 cubics. But since D 1 P 3 is a general complete intersection of 3 cubics, it contains 7 general points. Consequently, C (D 1 P 3 ) does not lie on any cubics, and so satisfies maximal ran for cubics. If = 3 and (d, g) {(3, 0), (4, 1)}, then partition D 1 P 3 = A B into two sets of 4 points, and let Q be a general quadric containing A. Since any subset of 7 points of D 1 P 3 are general, Q is smooth and does not contain any point of B; moreover, A is a set of 4 general points on Q, while B is a set of 4 general points in P 3 (although not independently general from A!). We now specialize C to a curve of type (a, 2) on Q, where { 1 if (d, g) = (3, 0); a = 2 if (d, g) = (4, 1). The exact sequence of sheaves 0 I B P 3(1) I A B C P 3(3) I A Q (3 a, 1) 0 gives rise to the long exact sequence in cohomology H 1 (I B P 3(1)) H 1 (I A B C P 3(3)) H 1 (I A Q (3 a, 1)). Since A and B are general sets of 4 points in Q and P 3 respectively, dim H 0 (O P 3(1)) = 4 and dim H 0 (O Q (3 a, 1)) = 8 2a 4, while H 1 (O P 3(1)) = H 1 (O Q (3 a, 1)) = 0, we conclude that H 1 (I B P 3(1)) = H 1 (I A Q (3 a, 1)) = 0, which implies H 1 (I A B C P 3(3)) = 0 as desired. If = 4 and (d, g) = (8, 6), then partition D 1 P 3 = A B into two sets of 4 points, and let S be a general cubic containing A. As in the previous cases, S is smooth and does not contain any point of B; moreover, A is a set of 4 general points on S, while B is a set of 4 general points in P 3. Write S as the blowup of P 2 at six points, L for the pullbac of the class of a line in P 2 to S, and E 1, E 2,..., E 6 for the six exceptional divisors. By Lemma 9.3 of [14] plus results of [10], we may specialize C to a curve on S of class The exact sequence of sheaves 6L E 1 E 2 2E 3 2E 4 2E 5 2E 6. 0 I B P 3(1) I A B C P 3(3) I A S (3L 2E 1 2E 2 E 3 E 4 E 5 E 6 ) 0 gives rise to the long exact sequence in cohomology H 1 (I B P 3(1)) H 1 (I A B C P 3(4)) H 1 (I A S (6L 3E 1 3E 2 2E 3 2E 4 2E 5 2E 6 )). Since the cone of effective curves on S is spanned by the 27 lines, the Naai-Moishezon criterion implies 9L 4E 1 4E 2 3E 3 3E 4 3E 5 3E 6 is ample. Consequently, by Kodaira vanishing, 6L 3E 1 3E 2 2E 3 2E 4 2E 5 2E 6 has no higher cohomology. In particular, by the Riemann Roch theorem for surfaces, dim H 0 (O S (6L 3E 1 3E 2 2E 3 2E 4 2E 5 2E 6 )) = 4. Since A and B are general sets of 4 points in S and P 3 respectively, we conclude that H 1 (I B P 3(1)) = H 1 (I A Q (6L 3E 1 3E 2 2E 3 2E 4 2E 5 2E 6 )) = 0, which implies H 1 (I A B C P 3(4)) = 0 as desired, thus completing the inductive step. 15

16 4 Quadrics In this section, we prove Theorem 1.3 for quadrics ( = 2); this will serve as another base case for our larger inductive argument. As in the previous section, we suppose ɛ = 0 and see to show T satisfies maximal ran. Our argument here will also be by induction, this time on r, using a construction due to Ballico [2]. The base case of r = 3 was done in Section 3, and the case n = 0 was done by Ballico in [2], so we suppose for our inductive argument that r 4 and n 1. Write d and g for the degree and genus of C. If g d + 2, then C cannot be a degenerate rational curve, and so C is a BN-curve and ( ) r + 2 2d + 1 g 2d + 1 g (r 1)(g d 2) [(r + 1)d rg r(r + 1)] = r 2 + 3r 1. 2 In particular, by results of Ballico [2], C does not lie on any quadrics. In particular, we conclude that C (D 1 D 2 D n ) P r also does not lie on any quadrics, as desired. We thus suppose g d + 1 for the remainder of this section. For our inductive argument, we pic a hyperplane H P r+1, transverse to P r P r+1, and write Λ = P r H for the corresponding hyperplane in P r. If C is nonspecial, then we invoe Corollary 4.2 of [17] if C is a BN-curve (respectively application of an automorphism of projective space if C is a degenerate rational curve) to degenerate C to a general BN-curve (respectively general rational curve) C Λ. We also invoe Corollary 4.2 of [17] to degenerate each D i to D i D i where D i H and D i is transverse to H, such that deg D i = r + 1. Note that (D 1 D n) P r is a collection of r + 1 general points, since Aut P r acts transitively on collections of r + 1 points in linear general position. Proposition 2.3 then implies Theorem 1.3 for T, as desired. If C is special (so g + r d 1 0), then we invoe Corollary 4.2 of [17] to degenerate each D i to a general BN-curve Di H. Writing d = d r 1 and g = g r 1, we have g = [(r + 1)d rg r(r + 1)] + (r + 1)(g + r d 1) 0 ρ(d, g, r 1) = [(r + 1)d rg r(r + 1)] + (g + r d 1) 0. We may therefore let C Λ be a general BN-curve of degree d and genus g. Since Aut Λ acts transitively on collections of r + 1 points in linear general position, we may therefore construct a reducible curve C C P r, where C Λ P r is as above, C P r is a general BN-curve of degree r + 1 and genus 1 transverse to Λ, and C Λ = C C is a set of r + 1 general points in Λ. This curve has degree d + r + 1 = d and genus g + r + 1 = g, and is a BN-curve by Theorem 1.9 of [12] since our assumption that g d + 1 implies d + (r + 1) g + r; this curve is thus a specialization of C. With (, d, g, h, ɛ 0, ɛ 0) = (2, r + 1, 1, 0, 0, 0), we note that (1) is an equality. In particular, r, d, g, and h satisfy either I(r, 2, d, g, r + 1, 1, h, 0, 0, 0) or S(r, 2, d, g, r + 1, 1, h, 0, 0, 0). Applying Proposition 2.2 thus yields the desired result. 5 Curves of Extreme Degree: n = 0 In this section, we deal with the easy cases of Theorem 1.3 when n = 0; i.e. with those cases where the Brill Noether number is far from zero or the restriction map is far from being an isomorphism. As mentioned in the introduction, we may assume ɛ = 0. 16

17 Definition 5.1. We say that integers r,, d, and g satisfy U(r,, d, g) if ( ) r r + r + d 1 0 ( 1)d g ( ) r + r + 0 g + r d 1 0 (r + 1)d rg r(r + 1) 0. In this section, we will show that Theorem 1.3 holds unless r,, d, and g satisfy U(r,, d, g). Since ρ(d, g, r) = (r + 1)d rg r(r + 1) 0 by assumption, Ballico and Ellia have proven Theorem 1.3 when n = 0 in the case d g + r, and ( ) ( ) ( ) ( ) r + r + 1 r + r + 1 r r + = and r + are integers, it remains to prove Theorem 1.3 when either d r ( ) r + r + or ( 1)d + 1 g r + = ( r + Since we have already proven Theorem 1.3 for r = 3 and = 2, we suppose r 4 and 3. For this, it suffices by Proposition 2.2 to show that r,, d, and g satisfy I(r,, d, g, d, g, 0, 0, 0, 0) or S(r,, d, g, d, g, 0, 0, 0, 0). By assumption either (1) holds with, or (2) holds with. It thus remains to show that if (2) holds with >, then (1) holds with, or in other words that d > r ( ) r + r + ( 1)d + 1 g ( ) r + r +. So assume d r (r+ ) r Since 3 and r 4 by assumption, we obtain d r ( ) ( ) ( ) r + + r r 1 r = = 1 r 1 r 1 6 r r r + 1 r; in particular, C is a BN-curve, and so ρ(d, g, r) = (r + 1)d rg r(r + 1) 0. Thus, ( 1)d + 1 g ( 1)d + 1 g as desired. (r + 1)d rg r(r + 1) r r 2r 1 = d + r + 2 r ( ( ) r 2r 1 r r + r r + = ( ) r + r + + ( ) r + r +, ) + 1 (r 1)( 3) + (r 4) r r + 2 ( ) r + r ). + r + r2 1 r (33)

18 6 Curves of Extreme Degree: n > 0 In this section, we deal with the easy cases of Theorem 1.3 when n > 0; i.e. with those cases where the restriction map is far from being an isomorphism. Definition 6.1. We say that integers r,, d, g, h, and ɛ 0 satisfy V (r,, d, g, h, ɛ 0 ) if ( ) r + d + 1 g + h + ɛ 0 0 ( ) r r + r + d r r + 1 h + r 1 0 g 0 (r + 1)d rg r(r + 1) 0 h r 1 0. In this section, we show it suffices to verify Theorem 1.3 when r,, d, g, h, and ɛ 0 satisfy V (r,, d, g, h, ɛ 0 ). First, we note that if d + 1 g + h + ɛ 0 ( ) r+ < 0, then T satisfies maximal ran if and only if T {p} does where p is a general point. We may thus increase ɛ (and therefore h) until ( ) r + d + 1 g + h + ɛ 0 0. (34) By assumption we have g 0, and since n 1 we have h d 1 r +1. It thus suffices to verify Theorem 1.3 when V (r,, d, g, h, ɛ 0 ) is satisfied, or (34) holds and C is a degenerate rational curve, or (34) holds and r r + ( ) r + d r h + r 1 < 0. (35) r + 1 We first verify Theorem 1.3 when (34) and (35) hold. Fix a hyperplane H P r. Applying Proposition 2.9, we may degenerate each D i to curves D i, and the points {p ɛ0 +1, p ɛ0 +2,..., p ɛ } to points {p ɛ 0 +1, p ɛ 0 +2,..., p ɛ}, with so that are sets of general points, and #({p ɛ 0 +1, p ɛ 0 +2,..., p ɛ} H) + #(D i H) = h h, {p ɛ 0 +1, p ɛ 0 +2,..., p ɛ} H H and {p ɛ 0 +1, p ɛ 0 +2,..., p ɛ} P r H P r D 1 H, D 2 H,..., D n H H and D 1 P r H, D 2 P r H,..., D n P r H P r are sets of subsets of hyperplane sections of independantly general BN-curves, provided that r, h, and h satisfy J(r, h, h ). Moreover, we can assume the second of these sets is a set of subsets 18

19 of hyperplane sections of independantly general nonspecial BN-curves provided that r, h, and h satisfy N(r, h, h ). Applying Proposition 2.2, it this remains to show there exists an integer h so that r,, d, g, h, and h satisfy { I(r,, d, g, d, g, h, h N(r, h, h ) if = 3;, ɛ 0, ɛ 0 ) and J(r, h, h ) otherwise. Since these upper and lower bounds on h are all integers, we just have to chec: ( ) r + r + ( 1)d 1 + g ɛ 0 d + h r ( ) r + r + ; { h h r h if = 3, 2r+2 r + r + 1 h otherwise; ( ) { r + r + h r h if = 3, 2r+2 ( 1)d 1 + g ɛ 0 h otherwise; h r + d + h r ( ) r + r + 1 r +. The first of these is just (34) upon rearrangement. The second is immediate if h 2r + 2, which implies r + h h r h h. And when h < 2r + 2, the second follows from our r+1 2r+2 assumption that h r 1 0, which implies r + h h = h r h r+1 2r+2. The final of these inequalities follows from (35), which implies r + h < d + h r (r+ ) r+1 r It thus remains to show the third of these inequalities. By separately considering the cases when C is a degenerate rational curve and when C is a BN-curve, we always have (r + 1)d rg (r + 1) 0. Combining this with (34), we conclude r 2r 1 r r 1 ( d + 1 g + h + ɛ 0 ( )) r + + (r + 1)d rg (r + 1) r r 1 0; or upon rearrangement, r 2r 1 r r 1 h ( ) r + r + + ( 1)d + 1 g + ɛ 0 rɛ r 0 + 2r (r 2r 1) r r 1 0. (r+ r+ ) It thus remains to show { r 2r 1 r r 1 h h r h 2r+2 h if = 3; otherwise. 19

20 Since h r h 2r+2 r+2 2r+2 h, this reduces in turn to r 2r 1 r r 1 { r+2 2r+2 if = 3; 1 otherwise. This is clear, this completing the verification of Theorem 1.3 when (34) and (35) hold. It thus remains to verify Theorem 1.3 when (34) holds and C is a degenerate rational curve, but (35) does not hold. First, we claim this implies = 3. Indeed, if C is a degenerate rational curve, then in particular g = 0; since ɛ 0 2, the condition (34) gives ( ) r + h d + 1. Since d r 1, this yields d + We conclude ( ) r r + r + d r r + 1 h d + = r r + 1 r (( ) r + r + 1 ( ) r + r ( ) r + r + 1 = r ( ) r + r + 1 > r r + 1 r r + 1 h + r 1 < ( ) r + ) d + 1 r r 1 r + 1 r r 1 r + 1 r2 r 2 r + 1 r + 1 d r r r + 1 r2 r 2 r + 2. r + 1 (r 1) r r r + 1 r ( ) r + r + r ( ) r + r r2 r + 1 r + 1 ( ( )) 1 r + = (r + 1)(r + ) (r + )(r 2 r + 1) (r r) ( ( )) 1 r + (r + 1)(r + ) (r + )(r 2 r + 1) (r r) 4 1 ( = 24(r + 1) ( 4) 4 (r 4) + 3( 4) 3 (r 4) 2 + 3( 4) 2 (r 4) 3 + ( 4)(r 4) 4 + 4( 4) ( 4) 3 (r 4) + 57( 4) 2 (r 4) ( 4)(r 4) 3 + 3(r 4) ( 4) ( 4) 2 (r 4) + 299( 4)(r 4) (r 4) ( 4) ( 4)(r 4) + 441(r 4) ( 4) ) (r 4) , 20

21 and so (35) holds if 4. Next we suppose = 3. In this case, ɛ 0 = 0, and so (34) yields We conclude Consequently, r r + 3 h r3 + 6r r 6 3d. d + r r + 1 h d + r ( ) r 3 r r r 3d 6 = r4 + 6r r 2 2r 1 6r + 6 r + 1 d r4 + 6r r 2 2r 1 (r 1) 6r + 6 r + 1 = r4 + 6r 3 r r 6. 6r + 6 ( ) r + 3 d 3 r 2)(r 4) h + r 1 r(r r + 1 3r Thus, (35) holds unless all of the above inequalities are equalities; as r 4 this forces r = 4 and d = 3 and h = 25. It thus remains to consider the case (r,, d, g, h) = (4, 3, 3, 0, 25). In this case, we let H P 3 P 4 be the hyperplane containing C. Applying Proposition 2.9, we may degenerate each D i to curves D i, and the points {p 1, p 2,..., p ɛ } to points {p 1, p 2,..., p ɛ}, with so that are sets of general points, and #({p 1, p 2,..., p ɛ} H) + #(D i H) = 15, {p 1, p 2,..., p ɛ} H H and {p 1, p 2,..., p ɛ} P 4 H P 4 D 1 H, D 2 H,..., D n H H and D 1 P 4 H, D 2 P 4 H,..., D n P 4 H P 4 are sets of subsets of hyperplane sections of independantly general nonspecial BN-curves. Applying Proposition 2.3 then yields the desired result. 7 The Inductive Argument In this section, we give our inductive argument to prove Theorem 1.3. We begin with the case of cubic polynomials ( = 3) with n = 0. In this case, combining Propositions 2.2, 2.11, and 2.13 (with t = 0), we see that it suffices to chec: 21

22 Lemma 7.1. Let r 4, and d and g be integers satisfying U(r, 3, d, g). Then either: There exist integers d and n satisfying X(r, d, g, d, n) and I(r, 3, d, g, d, 0, 0, 0, 0, 0); There exist integers d and n satisfying X(r, d, g, d, n) and S(r, 3, d, g, d, 0, 0, 0, 0, 0); There exist integers d, g, and n sastisfying Y (r, d, g, d, g, n, 0) and I(r, 3, d, g, d, g, 0, 0, 0, 0); or There exist integers d, g, and n sastisfying Y (r, d, g, d, g, n, 0) and S(r, 3, d, g, d, g, 0, 0, 0, 0). Proof. This will be deferred to Appendix B.1. Next, we consider the case of cubic polynomials with n > 0. In this case, combining Propositions 2.2, 2.9, 2.11, and 2.13 (with t = 0), we see that it suffices to chec: Lemma 7.2. Let r 4, and d, g, and h be integers satisfying V (r, 3, d, g, h, 0). Then either: There exist integers d, h, and n satisfying X(r, d, g, d, n), I(r, 3, d, g, d, 0, h, h, 0, 0), and K(r, h, h ); There exist integers d, h, and n satisfying X(r, d, g, d, n), I(r, 3, d, g, d, 0, h, h, 0, 0), and M(r, h, h ); There exist integers d, h, and n satisfying X(r, d, g, d, n), I(r, 3, d, g, d, 0, h, h, 0, 0), and N(r, h, h ); There exist integers d, g, h, and n satisfying Y (r, d, g, d, g, n, 0), I(r, 3, d, g, d, g, h, h, 0, 0), and K(r, h, h ); There exist integers d, g, h, and n satisfying Y (r, d, g, d, g, n, 0), I(r, 3, d, g, d, g, h, h, 0, 0), and L(r, h, h ); or 22

23 There exist integers d, g, h, and n satisfying Y (r, d, g, d, g, n, 0), I(r, 3, d, g, d, g, h, h, 0, 0), and N(r, h, h ). Proof. This will be deferred to Appendix B.2. Next, we consider the case of quartic polynomials ( = 4) with n = 0. In this case, combining Propositions 2.2, 2.13 (with t = 0), and 2.16 (again with t = 0), we see that it suffices to chec: Lemma 7.3. Let r 4, and d and g be integers satisfying U(r, 4, d, g). Then either: There exist integers d, g, and n sastisfying Y (r, d, g, d, g, n, 0) and I(r, 4, d, g, d, g, 0, 0, 0, 0); There exist integers d, g, and n sastisfying Y (r, d, g, d, g, n, 0) and S(r, 4, d, g, d, g, 0, 0, 0, 0); There exist integers d, g, m, and n sastisfying or Z(r, d, g, d, g, m, n, 0) and I(r, 4, d, g, d, g, 0, 0, 0, 0); There exist integers d, g, m, and n sastisfying Z(r, d, g, d, g, m, n, 0) and S(r, 4, d, g, d, g, 0, 0, 0, 0). Proof. This will be deferred to Appendix B.3. Next, we consider the case of quartic polynomials with n > 0. In this case, combining Propositions 2.2, 2.9, 2.11, 2.13 (with t = 0), and 2.16 (again with t = 0), we see that it suffices to chec: Lemma 7.4. Let r 4, and d, g, and h be integers satisfying V (r, 4, d, g, h, 0). Then either: There exist integers d, h, and n satisfying X(r, d, g, d, n), I(r, 4, d, g, d, 0, h, h, 0, 0), and J(r, h, h ); There exist integers d, g, h, and n satisfying Y (r, d, g, d, g, n, 0), I(r, 4, d, g, d, g, h, h, 0, 0), and J(r, h, h ); There exist integers d, g, h, and n satisfying Y (r, d, g, d, g, n, 0), I(r, 4, d, g, d, g, h, h, 0, 0), and K(r, h, h ); 23

24 There exist integers d, g, h, m, and n satisfying or Z(r, d, g, d, g, m, n, 0), I(r, 4, d, g, d, g, h, h, 0, 0), and J(r, h, h ); There exist integers d, g, h, m, and n satisfying Z(r, d, g, d, g, m, n, 0), I(r, 4, d, g, d, g, h, h, 0, 0), and K(r, h, h ). Proof. This will be deferred to Appendix B.4. Next, we consider the case of polynomials of higher degree ( 5) with n = 0. In this case, combining Propositions 2.2, 2.13 (with t = 2), and 2.16 (again with t = 2), we see that it suffices to chec: Lemma 7.5. Let r 4 and 5, and d and g be integers satisfying U(r,, d, g). Then either: There exist integers d, g, and n sastisfying Y (r, d, g, d, g, n, 2) and I(r,, d, g, d, g, 0, 0, 0, 0); There exist integers d, g, and n sastisfying Y (r, d, g, d, g, n, 2) and S(r,, d, g, d, g, 0, 0, 0, 0); There exist integers d, g, m, and n sastisfying or Z(r, d, g, d, g, m, n, 2) and I(r,, d, g, d, g, 0, 0, 0, 0); There exist integers d, g, m, and n sastisfying Z(r, d, g, d, g, m, n, 2) and S(r,, d, g, d, g, 0, 0, 0, 0). Proof. This will be deferred to Appendix B.5. Finally, we consider the case of polynomials of higher degree with n > 0. Combining Propositions 2.2, 2.9, 2.13 (with t = 2), and 2.16 (with t = 2), we see that it suffices to chec: Lemma 7.6. Let r 4 and 5, and d, g, and h be integers satisfying V (r,, d, g, h, 2). Then either: There exist integers d, g, h, n, and ɛ 0 satisfying Y (r, d, g, d, g, n, 2), I(r,, d, g, d, g, h, h, 2, ɛ 0), E( 2, ɛ 0, 3), and J(r, h, h ); 24

25 There exist integers d, g, h, m, n, and ɛ 0 satisfying Z(r, d, g, d, g, m, n, 2), I(r,, d, g, d, g, h, h, 2), E( 2, ɛ 0, 3), and J(r, h, h ); For every integer h satisfying A(r, h, h ), there exist integers d, g, n, and ɛ 0 satisfying or Y (r, d, g, d, g, n, 2), I(r,, d, g, d, g, h, h, 2), and E( 2, ɛ 0, 0); For every integer h satisfying A(r, h, h ), there exist integers d, g, m, n, and ɛ 0, satisfying Z(r, d, g, d, g, m, n, 2), I(r,, d, g, d, g, h, h, 2), and E( 2, ɛ 0, 0). Proof. This will be deferred to Appendix B.6. A Inequalities from Section 2 In this appendix, we give code in sage to chec or create algebraic expressions for all inequalities that appear in Section 2. # File name : inequalities. py def I(r,, d, g, dp, gp, h, hp, e0, e0p, B): # We pass B = binomial ( r +, ) as an extra argument. return [ ( - 1) * dp gp + hp + e0p - * B / (r + ), * d - g - ( - 1) * dp + gp + h - hp + e0 - e0p - r * B / ( r + ) ] def S(r,, d, g, dp, gp, h, hp, e0, e0p, B): # We pass B = binomial ( r +, ) as an extra argument. return [ * B / (r + ) - (( - 1) * dp gp + hp + e0p ), r * B / (r + ) - ( * d - g - ( - 1) * dp + gp + h - hp + e0 - e0p ) ] def E(e0, e0p, t): return [ e0p - t, e0 - e0p ] def A(r, h, hp): return [ hp, 25

26 ] h - (r + 1) * hp def J(r, h, hp): return [ hp - r - h / (r + 1), h - hp ] def K(r, h, hp): return [ 2 * r h, hp, h - hp ] def L(r, h, hp): return [ 3 * r h, hp - 2, 2 - hp ] def M(r, h, hp): return [ 3 * r h, hp - r - 2, h - hp - r/2 ] def N(r, h, hp): return [ hp - r - h / (r + 1), h - hp - r * h / (2 * r + 2) ] def Np(r, h, hp): # When 3 r + 3 <= h <= 4 r + 3: # then floor (h / (r + 1)) = 3 and floor (h / (2r + 2)) = 1, # so N(r, h, hp) is satisfied if: return [ h - 3 * r - 3, 4 * r h, hp - r - 3, h - hp - r ] 26

27 def X(r, d, g, dp, n): return [ g - n + 1, r * (d - dp) - (r - 1) * g + (r - 1) * n - r **2 + 1, n - 1, dp - n, r n, 2 * n + d - dp - g - r - 1 ] def Y(r, d, g, dp, gp, n, t): return [ gp, (r + 1) * dp - r * gp - r **2 - r, (2 * r - 3) * dp - (r - 2) **2 * gp - 2 * r **2 + 3 * r - 9, g - gp - n + 1, r * (d - dp) - (r - 1) * (g - gp) + (r - 1) * n - r **2 + 1, n - 1, dp - n - t, r * (d - dp) - (r - 4) * (g - gp) - 2 * n - 2 * r + 2, 2 * n + d + gp - dp - g - r - 2 ] def Z(r, d, g, dp, gp, m, n, t): return [ gp - (r + 1) * m, (r + 1) * dp - r * gp - r **2 - r, (2 * r - 3) * (dp - r * m) - (r - 2) **2 * (gp - (r + 1) * m) - 2 * r **2 + 3 * r - 9, g - gp - n + 1, r * (d - dp) - (r - 1) * (g - gp) + (r - 1) * n - r **2 + 1, n - 1, (dp - r * m) - n - t, r * (d - dp) - (r - 4) * (g - gp) - 2 * n - 2 * r + 2, 2 * n + d + gp - dp - g - r - 2, 2 * (dp - r * m) - (r - 3) * (gp - (r + 1) * m - 1) - (r - 1) * (r + 2), m ] def U(r,, d, g, B): # We pass B = binomial ( r +, ) as an extra argument. return [ r * B / (r + ) - d - 1, ( - 1) * d - g - * B / (r + ), g + r - d - 1, (r + 1) * d - r * g - r * (r + 1) ] def V(r,, d, g, h, e0, B): # We pass B = binomial ( r +, ) as an extra argument. 27

Signature redacted Signature of Author: _ Department of Mathematics April 5, 2018

Signature redacted Signature of Author: _ Department of Mathematics April 5, 2018 I The Maximal Rank Conjecture by Eric Kerner Larson A.B. Mathematics Harvard University, 2013 SUBMITTED TO THE DEPARTMENT OF MATHEMATICS IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR

More information

Interpolation with Bounded Error

Interpolation with Bounded Error Interpolation with Bounded Error Eric Larson arxiv:1711.01729v1 [math.ag] 6 Nov 2017 Abstract Given n general points p 1,p 2,...,p n P r it is natural to ask whether there is a curve of given degree d

More information

BRILL-NOETHER THEORY. This article follows the paper of Griffiths and Harris, "On the variety of special linear systems on a general algebraic curve.

BRILL-NOETHER THEORY. This article follows the paper of Griffiths and Harris, On the variety of special linear systems on a general algebraic curve. BRILL-NOETHER THEORY TONY FENG This article follows the paper of Griffiths and Harris, "On the variety of special linear systems on a general algebraic curve." 1. INTRODUCTION Brill-Noether theory is concerned

More information

LIMIT LINEAR SERIES AND THE MAXIMAL RANK CONJECTURE

LIMIT LINEAR SERIES AND THE MAXIMAL RANK CONJECTURE LIMIT LINEAR SERIES AND THE MAXIMAL RANK CONJECTURE BRIAN OSSERMAN Abstract. The maximal rank conjecture addresses the degrees of equations cutting out suitably general curves in projective spaces. We

More information

SPACES OF RATIONAL CURVES ON COMPLETE INTERSECTIONS

SPACES OF RATIONAL CURVES ON COMPLETE INTERSECTIONS SPACES OF RATIONAL CURVES ON COMPLETE INTERSECTIONS ROYA BEHESHTI AND N. MOHAN KUMAR Abstract. We prove that the space of smooth rational curves of degree e on a general complete intersection of multidegree

More information

SPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS

SPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS SPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS ROYA BEHESHTI AND N. MOHAN KUMAR Abstract. We prove that the space of smooth rational curves of degree e in a general complete intersection of multidegree

More information

INTERPOLATION PROBLEMS: DEL PEZZO SURFACES

INTERPOLATION PROBLEMS: DEL PEZZO SURFACES INTERPOLATION PROBLEMS: DEL PEZZO SURFACES AARON LANDESMAN AND ANAND PATEL Abstract. We consider the problem of interpolating projective varieties through points and linear spaces. After proving general

More information

BRILL-NOETHER THEORY, II

BRILL-NOETHER THEORY, II This article follows the paper of Griffiths and Harris, "On the variety of special linear systems on a general algebraic curve" 1 WARMUP ON DEGENERATIONS The classic first problem in Schubert calculus

More information

LINKED ALTERNATING FORMS AND LINKED SYMPLECTIC GRASSMANNIANS

LINKED ALTERNATING FORMS AND LINKED SYMPLECTIC GRASSMANNIANS LINKED ALTERNATING FORMS AND LINKED SYMPLECTIC GRASSMANNIANS BRIAN OSSERMAN AND MONTSERRAT TEIXIDOR I BIGAS Abstract. Motivated by applications to higher-rank Brill-Noether theory and the Bertram-Feinberg-Mukai

More information

COMPLEX ALGEBRAIC SURFACES CLASS 4

COMPLEX ALGEBRAIC SURFACES CLASS 4 COMPLEX ALGEBRAIC SURFACES CLASS 4 RAVI VAKIL CONTENTS 1. Serre duality and Riemann-Roch; back to curves 2 2. Applications of Riemann-Roch 2 2.1. Classification of genus 2 curves 3 2.2. A numerical criterion

More information

LINKED HOM SPACES BRIAN OSSERMAN

LINKED HOM SPACES BRIAN OSSERMAN LINKED HOM SPACES BRIAN OSSERMAN Abstract. In this note, we describe a theory of linked Hom spaces which complements that of linked Grassmannians. Given two chains of vector bundles linked by maps in both

More information

Normality of secant varieties

Normality of secant varieties Normality of secant varieties Brooke Ullery Joint Mathematics Meetings January 6, 2016 Brooke Ullery (Joint Mathematics Meetings) Normality of secant varieties January 6, 2016 1 / 11 Introduction Let X

More information

MODULI SPACES OF CURVES

MODULI SPACES OF CURVES MODULI SPACES OF CURVES SCOTT NOLLET Abstract. My goal is to introduce vocabulary and present examples that will help graduate students to better follow lectures at TAGS 2018. Assuming some background

More information

the complete linear series of D. Notice that D = PH 0 (X; O X (D)). Given any subvectorspace V H 0 (X; O X (D)) there is a rational map given by V : X

the complete linear series of D. Notice that D = PH 0 (X; O X (D)). Given any subvectorspace V H 0 (X; O X (D)) there is a rational map given by V : X 2. Preliminaries 2.1. Divisors and line bundles. Let X be an irreducible complex variety of dimension n. The group of k-cycles on X is Z k (X) = fz linear combinations of subvarieties of dimension kg:

More information

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37 FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37 RAVI VAKIL CONTENTS 1. Application of cohomology: Hilbert polynomials and functions, Riemann- Roch, degrees, and arithmetic genus 1 1. APPLICATION OF COHOMOLOGY:

More information

NUMERICAL MACAULIFICATION

NUMERICAL MACAULIFICATION NUMERICAL MACAULIFICATION JUAN MIGLIORE AND UWE NAGEL Abstract. An unpublished example due to Joe Harris from 1983 (or earlier) gave two smooth space curves with the same Hilbert function, but one of the

More information

Special determinants in higher-rank Brill-Noether theory

Special determinants in higher-rank Brill-Noether theory Special determinants in higher-rank Brill-Noether theory Brian Osserman University of California, Davis March 12, 2011 1 Classical Brill-Noether theory Let C be a smooth, projective curve of genus g over

More information

STABLE BASE LOCUS DECOMPOSITIONS OF KONTSEVICH MODULI SPACES

STABLE BASE LOCUS DECOMPOSITIONS OF KONTSEVICH MODULI SPACES STABLE BASE LOCUS DECOMPOSITIONS OF KONTSEVICH MODULI SPACES DAWEI CHEN AND IZZET COSKUN Contents 1. Introduction 1 2. Preliminary definitions and background 3 3. Degree two maps to Grassmannians 4 4.

More information

12. Linear systems Theorem Let X be a scheme over a ring A. (1) If φ: X P n A is an A-morphism then L = φ O P n

12. Linear systems Theorem Let X be a scheme over a ring A. (1) If φ: X P n A is an A-morphism then L = φ O P n 12. Linear systems Theorem 12.1. Let X be a scheme over a ring A. (1) If φ: X P n A is an A-morphism then L = φ O P n A (1) is an invertible sheaf on X, which is generated by the global sections s 0, s

More information

Rational curves on general type hypersurfaces

Rational curves on general type hypersurfaces Rational curves on general type hypersurfaces Eric Riedl and David Yang May 5, 01 Abstract We develop a technique that allows us to prove results about subvarieties of general type hypersurfaces. As an

More information

Non-uniruledness results for spaces of rational curves in hypersurfaces

Non-uniruledness results for spaces of rational curves in hypersurfaces Non-uniruledness results for spaces of rational curves in hypersurfaces Roya Beheshti Abstract We prove that the sweeping components of the space of smooth rational curves in a smooth hypersurface of degree

More information

Projective Schemes with Degenerate General Hyperplane Section II

Projective Schemes with Degenerate General Hyperplane Section II Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 44 (2003), No. 1, 111-126. Projective Schemes with Degenerate General Hyperplane Section II E. Ballico N. Chiarli S. Greco

More information

The Reducibility and Dimension of Hilbert Schemes of Complex Projective Curves

The Reducibility and Dimension of Hilbert Schemes of Complex Projective Curves The Reducibility and Dimension of Hilbert Schemes of Complex Projective Curves Krishna Dasaratha dasaratha@college.harvard.edu Advisor: Joe Harris Submitted to the Department of Mathematics in partial

More information

SESHADRI CONSTANTS ON SURFACES

SESHADRI CONSTANTS ON SURFACES SESHADRI CONSTANTS ON SURFACES KRISHNA HANUMANTHU 1. PRELIMINARIES By a surface, we mean a projective nonsingular variety of dimension over C. A curve C on a surface X is an effective divisor. The group

More information

PROBLEMS FOR VIASM MINICOURSE: GEOMETRY OF MODULI SPACES LAST UPDATED: DEC 25, 2013

PROBLEMS FOR VIASM MINICOURSE: GEOMETRY OF MODULI SPACES LAST UPDATED: DEC 25, 2013 PROBLEMS FOR VIASM MINICOURSE: GEOMETRY OF MODULI SPACES LAST UPDATED: DEC 25, 2013 1. Problems on moduli spaces The main text for this material is Harris & Morrison Moduli of curves. (There are djvu files

More information

Vector bundles in Algebraic Geometry Enrique Arrondo. 1. The notion of vector bundle

Vector bundles in Algebraic Geometry Enrique Arrondo. 1. The notion of vector bundle Vector bundles in Algebraic Geometry Enrique Arrondo Notes(* prepared for the First Summer School on Complex Geometry (Villarrica, Chile 7-9 December 2010 1 The notion of vector bundle In affine geometry,

More information

TWO LECTURES ON APOLARITY AND THE VARIETY OF SUMS OF POWERS

TWO LECTURES ON APOLARITY AND THE VARIETY OF SUMS OF POWERS TWO LECTURES ON APOLARITY AND THE VARIETY OF SUMS OF POWERS KRISTIAN RANESTAD (OSLO), LUKECIN, 5.-6.SEPT 2013 1. Apolarity, Artinian Gorenstein rings and Arithmetic Gorenstein Varieties 1.1. Motivating

More information

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 48

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 48 FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 48 RAVI VAKIL CONTENTS 1. A little more about cubic plane curves 1 2. Line bundles of degree 4, and Poncelet s Porism 1 3. Fun counterexamples using elliptic curves

More information

STABLE BASE LOCUS DECOMPOSITIONS OF KONTSEVICH MODULI SPACES

STABLE BASE LOCUS DECOMPOSITIONS OF KONTSEVICH MODULI SPACES STABLE BASE LOCUS DECOMPOSITIONS OF KONTSEVICH MODULI SPACES DAWEI CHEN AND IZZET COSKUN Abstract. In this paper, we determine the stable base locus decomposition of the Kontsevich moduli spaces of degree

More information

arxiv: v1 [math.ag] 28 Sep 2016

arxiv: v1 [math.ag] 28 Sep 2016 LEFSCHETZ CLASSES ON PROJECTIVE VARIETIES JUNE HUH AND BOTONG WANG arxiv:1609.08808v1 [math.ag] 28 Sep 2016 ABSTRACT. The Lefschetz algebra L X of a smooth complex projective variety X is the subalgebra

More information

RIMS-1743 K3 SURFACES OF GENUS SIXTEEN. Shigeru MUKAI. February 2012 RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES. KYOTO UNIVERSITY, Kyoto, Japan

RIMS-1743 K3 SURFACES OF GENUS SIXTEEN. Shigeru MUKAI. February 2012 RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES. KYOTO UNIVERSITY, Kyoto, Japan RIMS-1743 K3 SURFACES OF GENUS SIXTEEN By Shigeru MUKAI February 2012 RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES KYOTO UNIVERSITY, Kyoto, Japan K3 SURFACES OF GENUS SIXTEEN SHIGERU MUKAI Abstract. The

More information

Math 203A, Solution Set 6.

Math 203A, Solution Set 6. Math 203A, Solution Set 6. Problem 1. (Finite maps.) Let f 0,..., f m be homogeneous polynomials of degree d > 0 without common zeros on X P n. Show that gives a finite morphism onto its image. f : X P

More information

Complex Algebraic Geometry: Smooth Curves Aaron Bertram, First Steps Towards Classifying Curves. The Riemann-Roch Theorem is a powerful tool

Complex Algebraic Geometry: Smooth Curves Aaron Bertram, First Steps Towards Classifying Curves. The Riemann-Roch Theorem is a powerful tool Complex Algebraic Geometry: Smooth Curves Aaron Bertram, 2010 12. First Steps Towards Classifying Curves. The Riemann-Roch Theorem is a powerful tool for classifying smooth projective curves, i.e. giving

More information

div(f ) = D and deg(d) = deg(f ) = d i deg(f i ) (compare this with the definitions for smooth curves). Let:

div(f ) = D and deg(d) = deg(f ) = d i deg(f i ) (compare this with the definitions for smooth curves). Let: Algebraic Curves/Fall 015 Aaron Bertram 4. Projective Plane Curves are hypersurfaces in the plane CP. When nonsingular, they are Riemann surfaces, but we will also consider plane curves with singularities.

More information

LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL

LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL In this lecture we discuss a criterion for non-stable-rationality based on the decomposition of the diagonal in the Chow group. This criterion

More information

mult V f, where the sum ranges over prime divisor V X. We say that two divisors D 1 and D 2 are linearly equivalent, denoted by sending

mult V f, where the sum ranges over prime divisor V X. We say that two divisors D 1 and D 2 are linearly equivalent, denoted by sending 2. The canonical divisor In this section we will introduce one of the most important invariants in the birational classification of varieties. Definition 2.1. Let X be a normal quasi-projective variety

More information

THE EFFECTIVE CONE OF THE KONTSEVICH MODULI SPACE

THE EFFECTIVE CONE OF THE KONTSEVICH MODULI SPACE THE EFFECTIVE CONE OF THE KONTSEVICH MODULI SPACE IZZET COSKUN, JOE HARRIS, AND JASON STARR Abstract. In this paper we prove that the cone of effective divisors on the Kontsevich moduli spaces of stable

More information

DIVISORS ON NONSINGULAR CURVES

DIVISORS ON NONSINGULAR CURVES DIVISORS ON NONSINGULAR CURVES BRIAN OSSERMAN We now begin a closer study of the behavior of projective nonsingular curves, and morphisms between them, as well as to projective space. To this end, we introduce

More information

Segre classes of tautological bundles on Hilbert schemes of surfaces

Segre classes of tautological bundles on Hilbert schemes of surfaces Segre classes of tautological bundles on Hilbert schemes of surfaces Claire Voisin Abstract We first give an alternative proof, based on a simple geometric argument, of a result of Marian, Oprea and Pandharipande

More information

COMPLEX ALGEBRAIC SURFACES CLASS 9

COMPLEX ALGEBRAIC SURFACES CLASS 9 COMPLEX ALGEBRAIC SURFACES CLASS 9 RAVI VAKIL CONTENTS 1. Construction of Castelnuovo s contraction map 1 2. Ruled surfaces 3 (At the end of last lecture I discussed the Weak Factorization Theorem, Resolution

More information

INTERPOLATION ON SURFACES IN P 3

INTERPOLATION ON SURFACES IN P 3 INTERPOLATION ON SURFACES IN P 3 JACK HUIZENGA Abstract. Suppose S is a surface in P 3, and p 1,..., p r are general points on S. What is the dimension of the space of sections of O S (e) having singularities

More information

Stable maps and Quot schemes

Stable maps and Quot schemes Stable maps and Quot schemes Mihnea Popa and Mike Roth Contents 1. Introduction........................................ 1 2. Basic Setup........................................ 4 3. Dimension Estimates

More information

0.1 Complex Analogues 1

0.1 Complex Analogues 1 0.1 Complex Analogues 1 Abstract In complex geometry Kodaira s theorem tells us that on a Kähler manifold sufficiently high powers of positive line bundles admit global holomorphic sections. Donaldson

More information

12. Hilbert Polynomials and Bézout s Theorem

12. Hilbert Polynomials and Bézout s Theorem 12. Hilbert Polynomials and Bézout s Theorem 95 12. Hilbert Polynomials and Bézout s Theorem After our study of smooth cubic surfaces in the last chapter, let us now come back to the general theory of

More information

arxiv:math/ v1 [math.ag] 17 Oct 2006

arxiv:math/ v1 [math.ag] 17 Oct 2006 Remark on a conjecture of Mukai Arnaud BEAUVILLE Introduction arxiv:math/0610516v1 [math.ag] 17 Oct 2006 The conjecture mentioned in the title appears actually as a question in [M] (Problem 4.11): Conjecture.

More information

The Pennsylvania State University The Graduate School Eberly College of Science AN ASYMPTOTIC MUKAI MODEL OF M 6

The Pennsylvania State University The Graduate School Eberly College of Science AN ASYMPTOTIC MUKAI MODEL OF M 6 The Pennsylvania State University The Graduate School Eberly College of Science AN ASYMPTOTIC MUKAI MODEL OF M 6 A Dissertation in Mathematics by Evgeny Mayanskiy c 2013 Evgeny Mayanskiy Submitted in Partial

More information

EFFECTIVE CONES OF CYCLES ON BLOW-UPS OF PROJECTIVE SPACE

EFFECTIVE CONES OF CYCLES ON BLOW-UPS OF PROJECTIVE SPACE EFFECTIVE CONES OF CYCLES ON BLOW-UPS OF PROJECTIVE SPACE IZZET COSKUN, JOHN LESIEUTRE, AND JOHN CHRISTIAN OTTEM Abstract. In this paper, we study the cones of higher codimension (pseudo)effective cycles

More information

1 Basic Combinatorics

1 Basic Combinatorics 1 Basic Combinatorics 1.1 Sets and sequences Sets. A set is an unordered collection of distinct objects. The objects are called elements of the set. We use braces to denote a set, for example, the set

More information

Rational Curves on Hypersurfaces

Rational Curves on Hypersurfaces Rational Curves on Hypersurfaces The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Riedl, Eric. 015. Rational Curves on

More information

(1) is an invertible sheaf on X, which is generated by the global sections

(1) is an invertible sheaf on X, which is generated by the global sections 7. Linear systems First a word about the base scheme. We would lie to wor in enough generality to cover the general case. On the other hand, it taes some wor to state properly the general results if one

More information

IRREDUCIBILITY AND COMPONENTS RIGID IN MODULI OF THE HILBERT SCHEME OF SMOOTH CURVES

IRREDUCIBILITY AND COMPONENTS RIGID IN MODULI OF THE HILBERT SCHEME OF SMOOTH CURVES IRREDUCIBILITY AND COMPONENTS RIGID IN MODULI OF THE HILBERT SCHEME OF SMOOTH CURVES CHANGHO KEEM*, YUN-HWAN KIM AND ANGELO FELICE LOPEZ** Abstract. Denote by H d,g,r the Hilbert scheme of smooth curves,

More information

Vector Bundles on Algebraic Varieties

Vector Bundles on Algebraic Varieties Vector Bundles on Algebraic Varieties Aaron Pribadi December 14, 2010 Informally, a vector bundle associates a vector space with each point of another space. Vector bundles may be constructed over general

More information

2 G. D. DASKALOPOULOS AND R. A. WENTWORTH general, is not true. Thus, unlike the case of divisors, there are situations where k?1 0 and W k?1 = ;. r;d

2 G. D. DASKALOPOULOS AND R. A. WENTWORTH general, is not true. Thus, unlike the case of divisors, there are situations where k?1 0 and W k?1 = ;. r;d ON THE BRILL-NOETHER PROBLEM FOR VECTOR BUNDLES GEORGIOS D. DASKALOPOULOS AND RICHARD A. WENTWORTH Abstract. On an arbitrary compact Riemann surface, necessary and sucient conditions are found for the

More information

14. Rational maps It is often the case that we are given a variety X and a morphism defined on an open subset U of X. As open sets in the Zariski

14. Rational maps It is often the case that we are given a variety X and a morphism defined on an open subset U of X. As open sets in the Zariski 14. Rational maps It is often the case that we are given a variety X and a morphism defined on an open subset U of X. As open sets in the Zariski topology are very large, it is natural to view this as

More information

Introduction Curves Surfaces Curves on surfaces. Curves and surfaces. Ragni Piene Centre of Mathematics for Applications, University of Oslo, Norway

Introduction Curves Surfaces Curves on surfaces. Curves and surfaces. Ragni Piene Centre of Mathematics for Applications, University of Oslo, Norway Curves and surfaces Ragni Piene Centre of Mathematics for Applications, University of Oslo, Norway What is algebraic geometry? IMA, April 13, 2007 Outline Introduction Curves Surfaces Curves on surfaces

More information

If F is a divisor class on the blowing up X of P 2 at n 8 general points p 1,..., p n P 2,

If F is a divisor class on the blowing up X of P 2 at n 8 general points p 1,..., p n P 2, Proc. Amer. Math. Soc. 124, 727--733 (1996) Rational Surfaces with K 2 > 0 Brian Harbourne Department of Mathematics and Statistics University of Nebraska-Lincoln Lincoln, NE 68588-0323 email: bharbourne@unl.edu

More information

ON A THEOREM OF CAMPANA AND PĂUN

ON A THEOREM OF CAMPANA AND PĂUN ON A THEOREM OF CAMPANA AND PĂUN CHRISTIAN SCHNELL Abstract. Let X be a smooth projective variety over the complex numbers, and X a reduced divisor with normal crossings. We present a slightly simplified

More information

Pacific Journal of Mathematics

Pacific Journal of Mathematics Pacific Journal of Mathematics STABLE REFLEXIVE SHEAVES ON SMOOTH PROJECTIVE 3-FOLDS PETER VERMEIRE Volume 219 No. 2 April 2005 PACIFIC JOURNAL OF MATHEMATICS Vol. 219, No. 2, 2005 STABLE REFLEXIVE SHEAVES

More information

On the Waring problem for polynomial rings

On the Waring problem for polynomial rings On the Waring problem for polynomial rings Boris Shapiro jointly with Ralf Fröberg, Giorgio Ottaviani Université de Genève, March 21, 2016 Introduction In this lecture we discuss an analog of the classical

More information

ALGEBRAIC GEOMETRY I, FALL 2016.

ALGEBRAIC GEOMETRY I, FALL 2016. ALGEBRAIC GEOMETRY I, FALL 2016. DIVISORS. 1. Weil and Cartier divisors Let X be an algebraic variety. Define a Weil divisor on X as a formal (finite) linear combination of irreducible subvarieties of

More information

Algebraic Geometry. Question: What regular polygons can be inscribed in an ellipse?

Algebraic Geometry. Question: What regular polygons can be inscribed in an ellipse? Algebraic Geometry Question: What regular polygons can be inscribed in an ellipse? 1. Varieties, Ideals, Nullstellensatz Let K be a field. We shall work over K, meaning, our coefficients of polynomials

More information

arxiv:math/ v3 [math.ag] 27 Jan 2004

arxiv:math/ v3 [math.ag] 27 Jan 2004 arxiv:math/0303382v3 [math.ag] 27 Jan 2004 MORE ÉTALE COVERS OF AFFINE SPACES IN POSITIVE CHARACTERISTIC KIRAN S. KEDLAYA Abstract. We prove that every geometrically reduced projective variety of pure

More information

A NICE PROOF OF FARKAS LEMMA

A NICE PROOF OF FARKAS LEMMA A NICE PROOF OF FARKAS LEMMA DANIEL VICTOR TAUSK Abstract. The goal of this short note is to present a nice proof of Farkas Lemma which states that if C is the convex cone spanned by a finite set and if

More information

Under these assumptions show that if D X is a proper irreducible reduced curve with C K X = 0 then C is a smooth rational curve.

Under these assumptions show that if D X is a proper irreducible reduced curve with C K X = 0 then C is a smooth rational curve. Example sheet 3. Positivity in Algebraic Geometry, L18 Instructions: This is the third and last example sheet. More exercises may be added during this week. The final example class will be held on Thursday

More information

Algebraic Geometry Spring 2009

Algebraic Geometry Spring 2009 MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

More information

APPENDIX 3: AN OVERVIEW OF CHOW GROUPS

APPENDIX 3: AN OVERVIEW OF CHOW GROUPS APPENDIX 3: AN OVERVIEW OF CHOW GROUPS We review in this appendix some basic definitions and results that we need about Chow groups. For details and proofs we refer to [Ful98]. In particular, we discuss

More information

MODULI OF VECTOR BUNDLES ON CURVES AND GENERALIZED THETA DIVISORS

MODULI OF VECTOR BUNDLES ON CURVES AND GENERALIZED THETA DIVISORS MODULI OF VECTOR BUNDLES ON CURVES AND GENERALIZED THETA DIVISORS MIHNEA POPA 1. Lecture II: Moduli spaces and generalized theta divisors 1.1. The moduli space. Back to the boundedness problem: we want

More information

Welsh s problem on the number of bases of matroids

Welsh s problem on the number of bases of matroids Welsh s problem on the number of bases of matroids Edward S. T. Fan 1 and Tony W. H. Wong 2 1 Department of Mathematics, California Institute of Technology 2 Department of Mathematics, Kutztown University

More information

FANO MANIFOLDS AND BLOW-UPS OF LOW-DIMENSIONAL SUBVARIETIES. 1. Introduction

FANO MANIFOLDS AND BLOW-UPS OF LOW-DIMENSIONAL SUBVARIETIES. 1. Introduction FANO MANIFOLDS AND BLOW-UPS OF LOW-DIMENSIONAL SUBVARIETIES ELENA CHIERICI AND GIANLUCA OCCHETTA Abstract. We study Fano manifolds of pseudoindex greater than one and dimension greater than five, which

More information

Algebraic Geometry (Math 6130)

Algebraic Geometry (Math 6130) Algebraic Geometry (Math 6130) Utah/Fall 2016. 2. Projective Varieties. Classically, projective space was obtained by adding points at infinity to n. Here we start with projective space and remove a hyperplane,

More information

COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY

COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY BRIAN OSSERMAN Classical algebraic geometers studied algebraic varieties over the complex numbers. In this setting, they didn t have to worry about the Zariski

More information

MINIMAL FREE RESOLUTIONS OF GENERAL POINTS LYING ON CUBIC SURFACES IN P 3

MINIMAL FREE RESOLUTIONS OF GENERAL POINTS LYING ON CUBIC SURFACES IN P 3 MINIMAL FREE RESOLUTIONS OF GENERAL POINTS LYING ON CUBIC SURFACES IN P 3 JUAN MIGLIORE AND MEGAN PATNOTT 1. Introduction Since it was first stated by Lorenzini in [18], the Minimal Resolution Conjecture

More information

Lines on Projective Hypersurfaces

Lines on Projective Hypersurfaces Lines on Projective Hypersurfaces Roya Beheshti Abstract We study the Hilbert scheme of lines on hypersurfaces in the projective space. The main result is that for a smooth Fano hypersurface of degree

More information

Secant Varieties of Segre Varieties. M. Catalisano, A.V. Geramita, A. Gimigliano

Secant Varieties of Segre Varieties. M. Catalisano, A.V. Geramita, A. Gimigliano . Secant Varieties of Segre Varieties M. Catalisano, A.V. Geramita, A. Gimigliano 1 I. Introduction Let X P n be a reduced, irreducible, and nondegenerate projective variety. Definition: Let r n, then:

More information

The cocycle lattice of binary matroids

The cocycle lattice of binary matroids Published in: Europ. J. Comb. 14 (1993), 241 250. The cocycle lattice of binary matroids László Lovász Eötvös University, Budapest, Hungary, H-1088 Princeton University, Princeton, NJ 08544 Ákos Seress*

More information

Chern classes à la Grothendieck

Chern classes à la Grothendieck Chern classes à la Grothendieck Theo Raedschelders October 16, 2014 Abstract In this note we introduce Chern classes based on Grothendieck s 1958 paper [4]. His approach is completely formal and he deduces

More information

Curves of genus 2 on rational normal scrolls and scrollar syzygies. Andrea Hofmann

Curves of genus 2 on rational normal scrolls and scrollar syzygies. Andrea Hofmann Curves of genus on rational normal scrolls and scrollar syzygies Andrea Hofmann December 010 Acknowledgements First of all, I would like to thank my advisor Kristian Ranestad for interesting discussions

More information

Structure of elliptic curves and addition laws

Structure of elliptic curves and addition laws Structure of elliptic curves and addition laws David R. Kohel Institut de Mathématiques de Luminy Barcelona 9 September 2010 Elliptic curve models We are interested in explicit projective models of elliptic

More information

Punctual Hilbert Schemes of the Plane

Punctual Hilbert Schemes of the Plane Punctual Hilbert Schemes of the Plane An undergraduate thesis submitted by Andrew Gordon Advised by Joesph Harris March 19, 2018 Contents 1 Introduction 2 2 Punctual Hilbert Schemes of the Plane 2 2.1

More information

The Pfaffian-Grassmannian derived equivalence

The Pfaffian-Grassmannian derived equivalence The Pfaffian-Grassmannian derived equivalence Lev Borisov, Andrei Căldăraru Abstract We argue that there exists a derived equivalence between Calabi-Yau threefolds obtained by taking dual hyperplane sections

More information

a double cover branched along the smooth quadratic line complex

a double cover branched along the smooth quadratic line complex QUADRATIC LINE COMPLEXES OLIVIER DEBARRE Abstract. In this talk, a quadratic line complex is the intersection, in its Plücker embedding, of the Grassmannian of lines in an 4-dimensional projective space

More information

where m is the maximal ideal of O X,p. Note that m/m 2 is a vector space. Suppose that we are given a morphism

where m is the maximal ideal of O X,p. Note that m/m 2 is a vector space. Suppose that we are given a morphism 8. Smoothness and the Zariski tangent space We want to give an algebraic notion of the tangent space. In differential geometry, tangent vectors are equivalence classes of maps of intervals in R into the

More information

The geography of irregular surfaces

The geography of irregular surfaces Università di Pisa Classical Algebraic Geometry today M.S.R.I., 1/26 1/30 2008 Summary Surfaces of general type 1 Surfaces of general type 2 3 and irrational pencils Surface = smooth projective complex

More information

Oral exam practice problems: Algebraic Geometry

Oral exam practice problems: Algebraic Geometry Oral exam practice problems: Algebraic Geometry Alberto García Raboso TP1. Let Q 1 and Q 2 be the quadric hypersurfaces in P n given by the equations f 1 x 2 0 + + x 2 n = 0 f 2 a 0 x 2 0 + + a n x 2 n

More information

then D 1 D n = D 1 D n.

then D 1 D n = D 1 D n. Lecture 8. Intersection theory and ampleness: revisited. In this lecture, X will denote a proper irreducible variety over k = k, chark = 0, unless otherwise stated. We will indicate the dimension of X

More information

Corollary. Let X Y be a dominant map of varieties, with general fiber F. If Y and F are rationally connected, then X is.

Corollary. Let X Y be a dominant map of varieties, with general fiber F. If Y and F are rationally connected, then X is. 1 Theorem. Let π : X B be a proper morphism of varieties, with B a smooth curve. If the general fiber F of f is rationally connected, then f has a section. Corollary. Let X Y be a dominant map of varieties,

More information

k k would be reducible. But the zero locus of f in A n+1

k k would be reducible. But the zero locus of f in A n+1 Math 145. Bezout s Theorem Let be an algebraically closed field. The purpose of this handout is to prove Bezout s Theorem and some related facts of general interest in projective geometry that arise along

More information

The Cone Theorem. Stefano Filipazzi. February 10, 2016

The Cone Theorem. Stefano Filipazzi. February 10, 2016 The Cone Theorem Stefano Filipazzi February 10, 2016 These notes are supposed to be a handout for the student seminar in algebraic geometry at the University of Utah. In this seminar, we will give an overview

More information

A Note on Dormant Opers of Rank p 1 in Characteristic p

A Note on Dormant Opers of Rank p 1 in Characteristic p A Note on Dormant Opers of Rank p 1 in Characteristic p Yuichiro Hoshi May 2017 Abstract. In the present paper, we prove that the set of equivalence classes of dormant opers of rank p 1 over a projective

More information

Curves on P 1 P 1. Peter Bruin 16 November 2005

Curves on P 1 P 1. Peter Bruin 16 November 2005 Curves on P 1 P 1 Peter Bruin 16 November 2005 1. Introduction One of the exercises in last semester s Algebraic Geometry course went as follows: Exercise. Let be a field and Z = P 1 P 1. Show that the

More information

THE ENVELOPE OF LINES MEETING A FIXED LINE AND TANGENT TO TWO SPHERES

THE ENVELOPE OF LINES MEETING A FIXED LINE AND TANGENT TO TWO SPHERES 6 September 2004 THE ENVELOPE OF LINES MEETING A FIXED LINE AND TANGENT TO TWO SPHERES Abstract. We study the set of lines that meet a fixed line and are tangent to two spheres and classify the configurations

More information

arxiv: v1 [math.ag] 13 Mar 2019

arxiv: v1 [math.ag] 13 Mar 2019 THE CONSTRUCTION PROBLEM FOR HODGE NUMBERS MODULO AN INTEGER MATTHIAS PAULSEN AND STEFAN SCHREIEDER arxiv:1903.05430v1 [math.ag] 13 Mar 2019 Abstract. For any integer m 2 and any dimension n 1, we show

More information

CONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP

CONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP CONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP TOSHIHIRO SHODA Abstract. In this paper, we study a compact minimal surface in a 4-dimensional flat

More information

Special cubic fourfolds

Special cubic fourfolds Special cubic fourfolds 1 Hodge diamonds Let X be a cubic fourfold, h H 2 (X, Z) be the (Poincaré dual to the) hyperplane class. We have h 4 = deg(x) = 3. By the Lefschetz hyperplane theorem, one knows

More information

The Grothendieck Ring of Varieties

The Grothendieck Ring of Varieties The Grothendieck Ring of Varieties Ziwen Zhu University of Utah October 25, 2016 These are supposed to be the notes for a talk of the student seminar in algebraic geometry. In the talk, We will first define

More information

Betti numbers of abelian covers

Betti numbers of abelian covers Betti numbers of abelian covers Alex Suciu Northeastern University Geometry and Topology Seminar University of Wisconsin May 6, 2011 Alex Suciu (Northeastern University) Betti numbers of abelian covers

More information

P m 1 P(H 0 (X, O X (D)) ). Given a point x X, let

P m 1 P(H 0 (X, O X (D)) ). Given a point x X, let 3. Ample and Semiample We recall some very classical algebraic geometry. Let D be an integral Weil divisor. Provided h 0 (X, O X (D)) > 0, D defines a rational map: φ = φ D : X Y. The simplest way to define

More information

A CONJECTURE ON RATIONAL APPROXIMATIONS TO RATIONAL POINTS

A CONJECTURE ON RATIONAL APPROXIMATIONS TO RATIONAL POINTS A CONJECTURE ON RATIONAL APPROXIMATIONS TO RATIONAL POINTS DAVID MCKINNON Abstract. In this paper, we examine how well a rational point P on an algebraic variety X can be approximated by other rational

More information

On the Hilbert Functions of Disjoint Unions of a Linear Space and Many Lines in P n

On the Hilbert Functions of Disjoint Unions of a Linear Space and Many Lines in P n International Mathematical Forum, 5, 2010, no. 16, 787-798 On the Hilbert Functions of Disjoint Unions of a Linear Space and Many Lines in P n E. Ballico 1 Dept. of Mathematics University of Trento 38123

More information

TROPICAL BRILL-NOETHER THEORY

TROPICAL BRILL-NOETHER THEORY TROPICAL BRILL-NOETHER THEORY 10. Clifford s Theorem In this section we consider natural relations between the degree and rank of a divisor on a metric graph. Our primary reference is Yoav Len s Hyperelliptic

More information