Some general results on plane curves with total inflection points

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1 Some general results on plane curves with total inflection points Filip Cools and Marc Coppens October 19, 006 Abstract. In this paper we study plane curves of degree d with e total inflection points, for nonzero natural numbers d and e. MSC. 14H10, 14H50, 14N0. 0 Introduction Let e and d be nonzero natural numbers and denote by V d,e the set of elements ((L 1, P 1,..., (L e, P e with L 1,..., L e lines in the complex projective plane P, P i a point on the line L i for each i, P i L j for each i j and such that there exists a plane curve C of degree d with contact order d with L i at P i for each i. So P 1,..., P e are total inflection points of C. In [5], the case d 4 (i.e. quartic curves has been studied intensively. The main tool used in that thesis is the so-called λ-invariant, which is nothing else than a cross ratio of four points. In [3], the cases e 1, have been handled and also the cases e 3, 4 for some special configurations of the lines L i and the points P i. In Section 1 of this paper, we will first prove a general result (Proposition 1.3. Note that part (a of this Proposition is already known (see [5, Chapter II, Lemma.15] or [3, Theorem A]. In Section, we prove a generalization of the main result of a paper of E. Ballico (see []. In Section 3, we introduce the notion of expected dimension for a component of V d,e. In a following paper of the authors, the established techniques of this paper will be used to describe all the components of V d,e in case e is equal to 3 or K.U.Leuven, Department of Mathematics, Celestijnenlaan 00B, B-3001 Leuven, Belgium, Filip.Cools@wis.kuleuven.be. Katholieke Hogeschool Kempen, Departement Industrieel Ingenieur en Biotechniek, Kleinhoefstraat 4, B-440 Geel, Belgium, Marc.Coppens@khk.be; the author is affiliated with K.U.Leuven as Research Fellow. 1

2 4, together with their images in the moduli spaces. Both authors are partially supported by the Fund of Scientific Research - Flanders (G Definition and a general result Definition 1.1. Let P be the complex projective plane and P be the incidence relation in (P P, thus P (L, P P L}. If d and e are nonzero natural numbers, we denote by V d,e (P e the set of elements (L, P ((L 1, P 1,..., (L e, P e with P i L j for all i j (hence also L i L j for i j and such that there exists a plane curve Γ of degree d, not containing any of the lines L i, with i(l i.γ, P i d. We write V d,e to denote the closure of V d,e in (P e. Assume (L, P V d,e. On L i we have the divisor dp i, so we can consider the subscheme dp i P. Denote by dp the subscheme dp 1 + +dp e P of length ed. Denote by V (L, P Γ(P, O P (d the set s dp Z(s}, hence P(V (L, P is the associated linear system of plane curves. Remark 1.. We can easily see that V d,1 P and that dim(v (L, P ( d+ d for all (L, P P. This follows from the fact that for each line L (P the restriction map Γ(P, O P (d Γ(L, O L (d is surjective and dp L corresponds to a unique element of P(Γ(L, O L (d. Proposition 1.3. Let (L, P V d,e. (a dim(v (L, P ( ( d e+ + 1, where n is defined to be 0 if n <. (b Let L be a line in P with P i L for 1 i e. Let V L (L, P be the image of the restriction map V (L, P Γ(L, O L (d. Let P i0 L i L for 1 i e. If d e, dim(v L (L, P d e+ and P(V L (L, P is a linear system g d e+1 d on L containing P P e0 + g d e d e. If d < e, dim(v L(L, P 1. (c Under the assumptions of (b, for P L with P P i0 for all 1 i e one has ((L, P, (L, P V d,e+1 if and only if dp P(V L (L, P. Proof: Note that (a is true in case e 1 by Remark 1.. Consider (L 1, P 1 P and let L (P be a line different from L 1. Let l (resp. l 1 be an equation of L (resp. L 1 and let f Ker[V (L 1, P 1 Γ(L, O L (d], hence l divides f. The divisor dp 1 + P 10 L 1 belongs to Z(f, hence l 1 divides f. This proves ll 1 Γ(P, O P (d if d, Ker[V (L 1, P 1 Γ(L, O L (d] 0 if d 1,

3 hence the dimension of Ker[V (L 1, P 1 Γ(L, O L (d] is equal to ( d and ( d dim(v L (L 1, P 1 dim(v (L 1, P 1 d + 1. This proves V L (L 1, P 1 Γ(L, O L (d, which is (b in case e 1. For P L with P P 10 the divisor dp L corresponds to a unique element of P(V L (L 1, P 1, hence ((L 1, P 1, (L, P V d, proving (c in case e 1, and we find dim(v ((L 1, P 1, (L, P dim(v L (L 1, P 1 d ( d ((a in case e. Take e and assume that (a is true for e. Let l i (resp. l be an equation for L i (resp. L. As before we find ll 1... l e Γ(P, O P (d e 1 if d e, Ker[V (L, P Γ(L, O L (d] 0 if d < e, hence the dimension of Ker[V (L, P Γ(L, O L (d] is equal to ( d e+1 and ( d e + 1 dim(v L (L, P dim(v (L, P ( ( d e + d e + 1 d e + if d e, if d < e. If d e, l 1... l e Γ(P, O P (d e V (L, P and we obtain that P P e0 + g d e d e P(V L(L, P. This finishes the proof of (b for e. Moreover, claim (c holds because of the construction of P(V L (L, P. In particular, if (L, P : ((L, P, (L, P V d,e+1, then dp corresponds to a unique divisor of P(V L (L, P. This implies dim(v (L, P dim(v (L, P dim(p(v L (L, P ( d e + d e + 1 if d e if d < e ( d (e , hence (a holds for e + 1. Remark 1.4. From the proof of Proposition 1.3 in case e we also obtain V d, ((L 1, P 1, (L, P P i L j for i j}, hence V d, (P. 3

4 A generalization of a result of Ballico In this section, we will prove the following generalization of a result of E. Ballico (see [], where the case e has been handled. The main reason why E. Ballico restricted himself to the case e, is that the analogous statement of Remark 1.4 does not hold for e 3. Due to the geometrical clear statements of Proposition 1.3, we are able to make the natural generalization in case e 3. Proposition.1. Assume that (L, P V d,e with d e and that z is an integer satisfying 3z < ( ( d e+ and z d e 1. Let O O1,..., O z } be general points in P and let V (L, P, O s V (L, P Z(s is singular at O 1,..., O z }. (a dim(v (L, P, O dim(v (L, P 3z and a general element of V (L, P, O does not contain some line L i for 1 i e. (b If (d e, z (6, 9, then a general element of V (L, P, O is smooth outside O 1,..., O z and has ordinary nodes at O 1,..., O z. To prove Proposition.1, we use the following classical result of E. Arbarello and M. Cornalba (see [1]. Proposition.. Let V f (O s Γ(P, O P (f Z(s singular at O 1,..., O z }. 1. If 3z < ( ( f+ and z f 1, then dim(vf (O ( f+ 3z.. In case (f, z (6, 9, a general element of V f (O corresponds to an irreducible curve Γ smooth outside O 1,..., O z and having an ordinary node at O 1,..., O z. Proof of Proposition.1: Let (L x, P x ((L 1, P 1,..., (L x, P x for all 1 x e, V (L 0, P 0 Γ(P, O P (d and V (L 0, P 0, O V d (O. Claim i : For 0 x e we have dim(v (L x, P x, O dim(v (L x, P x 3z and a general element of P(V (L x, P x, O does not contain any of the lines L i (for 1 i x. Proof of Claim i : Since 3z < ( ( d e+ d+ ( and z d e 1 ( d 1, this claim holds in case x 0 (classical case. Assume x 1 and that the claim holds for x 1. Clearly Ker[V (L x 1, P x 1, O Γ(L x, O Lx (d] l 1 l x V d x (O. Since 1 x e, one has 3z < ( ( d x+ and z d x 1, hence the dimension of that kernel is equal to ( d x+ 3z, because of Proposition.. 4

5 Let V Lx (L x 1, P x 1, O Im[V (L x 1, P x 1, O Γ(L x, O Lx (d], then we find by using Proposition 1.3 (a that ( d x + dim(v Lx (L x 1, P x 1, O dim(v (L x 1, P x 1, O + 3z ( d x + dim(v (L x 1, P x 1 ( ( d (x 1 + d x δ 0,x 1 } d x + 3 δ 0,x 1 dim(v Lx (L x 1, P x 1, where δ 0,x 1 is a Kronecker delta. This implies that V Lx (L x 1, P x 1, O V Lx (L x 1, P x 1 since V Lx (L x 1, P x 1, O V Lx (L x 1, P x 1. So from (L x, P x V d,x it follows that dp x P(V Lx (L x 1, P x 1 P(V Lx (L x 1, P x 1, O, hence dim(v (L x, P x, O dim(v (L x 1, P x 1, O (d x + δ 0,x 1 dim(v (L x 1, P x 1 (d x + δ 0,x 1 3z dim(v (L x, P x 3z and a general element of P(V (L x, P x, O does not contain L x. If it would contain L i for some 1 i x 1 then it should contain the divisor P ix + dp x L x (here P ix L i L x, hence it would contain L x, a contradiction. This finishes the proof of Claim i. Since (d e, z (6, 9, there exists an irreducible curve Γ 0 V d e (O smooth outside O 1,..., O z having ordinary nodes at O 1,..., O z. Claim ii : We can assume that P i Γ 0 for 1 i e. Proof of Claim ii : Varying O 1,..., O z, the closure of the union of the loci V d e (O contains the union of d e general lines in P (see for example [4]. Such union does not contain any of those points P i, hence for O 1,..., O z general we can assume P i Γ 0 for 1 i e. It follows that the curve L L e +Γ 0 P(V (L, P is smooth at P 1,..., P e. From Claim i we know that Γ P(V (L, P, O general does not contain any of the lines L i (1 i e hence Γ L i P i }. It follows that the fixed locus of P(V (L, P, O is contained in the finite set P 1,..., P e } (Γ Γ 0. 5

6 A general element of P(V (L, P, O is singular at one of the points if and only if each element of P(V (L, P, O is singular at that point. We already knew this does not hold for P 1,..., P e. Also, since Γ 0 is smooth outside O 1,..., O z and Γ Γ 0 L i, it follows that a general element of P(V (L, P, O is smooth outside O 1,..., O z. Since L L e + Γ 0 has ordinary nodes at O 1,..., O z, a general element Γ of P(V (L, P, O has ordinary nodes at O 1,..., O z. 3 Expected dimension of a component of V d,e Let (P e,0 be the open subset of (P e of elements (L, P satisfying P i L j for i j. For (L, P (P e,0 and i j we write P i,j L i L j and we write g i to denote the linear system P 1,i P i 1,i + g d i+1 d i+1 on L i in case i d + 1. We take g i if i > d + 1. Assume e 3 and let G e be the space of pairs (g, (L, P with (L, P (P e,0 and g being an (e -tuple (g 3,..., g e of linear systems as follows. For 3 i e, g i is a linear system g maxd i+,0} d on L i containing g i. Let τ : G e (P e,0 be the natural projection. It is a smooth morphism of relative dimension e i1 i (e 1(e. Claim 3.1: There exist sections S 1 and S of τ such that V d,e τ(s 1 S. Corollary 3.. Each component of V d,e has codimension at most (e 1(e inside (P e. Definition 3.3. Let V be a component of V d,e. We say V has the expected dimension if dim(v 3e (e 1(e and we say that V is of exceptional dimension if dim(v > 3e (e 1(e. In case e 9 (this is a bound independent of d then all components of V d,e have exceptional dimension. Proof of Claim 3.1: The first section S 1 is defined by S 1 (L, P g with g i g i dp i } gd d on L i. In order to define S (L, P h, we are going to make a well-defined construction imitating the construction of the linear space V (L x, P x for 1 x e. We are going to define h inductively starting with h 3 P(V L3 (L, P. Inductively we also are going to define a sequence of subspaces V (L e, P e V (L e 1, P e 1 V (L 3, P 3 V (L, P such that the dimension of V (L x, P x is equal to ( d x+ +1, the space V (L x, P x contains l 1... l x Γ(P, O P (d x if x d and if s V (L x, P x and i 6

7 minx, d + 1}, then either l i Z(s or the multiplicity of Z(s L i at P i is more then d i + 1. Let D 3 h 3 be the divisor in h 3 having maximal multiplicity at P 3 (being at least d 1 and let W 3 V L3 (L, P be the corresponding 1-dimensional linear subspace. Then V (L 3, P 3 is the inverse image of W 3 in V (L, P. Since l 1 l l 3 Γ(P, O P (d 3 if d 3, Ker[V (L, P V L3 (L, P ] 0 if d < 3, one ( has l 1 l l 3 Γ(P, O P (d 3 V (L 3, P 3 if d 3 and dim(v (L 3, P 3 d Now let 3 < x e and assume h x 1 and V (L x 1, P x 1 are defined. Let s Ker[V (L x 1, P x 1 Γ(L x, O Lx (d], hence l x divides s. We are going to prove that l i divides s for 1 i x 1 too. Since Z(s contains P 1,x + dp 1 L 1 we find l 1 divides s. Let 1 < i 0 x 1 and assume l i divides s for 1 i < i 0. If i 0 d +, l 1... l i0 1 divides s, so s 0 and a fortiori l i0 divides s. If i 0 d + 1, Z(s contains P 1,i P i0 1,i 0 L i0. If l i0 does not divide s then Z(s L i0 is a divisor with multiplicity at most d i at P i0, a contradiction. Hence l i0 divides s. Since l 1... l x 1 Γ(P, O P (d x + 1 V (L x 1, P x 1 if x d + 1, we find Ker[V l 1... l x Γ(P, O P (d x if x d, (L x 1, P x 1 Γ(L x, O Lx (d] 0 if x > d. Let V L x (L x 1, P x 1 Im[V (L x 1, P x 1 Γ(L x, O Lx (d] then we find ( d x + dim(v L x (L x 1, P x 1 dim(v (L x 1, P x 1 d x + 3 if x d + 1, 1 if x > d + 1. We define h x P(V L x (L x 1, P x 1. Since l 1... l x 1 Γ(P, O P (d x + 1 V (L x 1, P x 1 if x d + 1, we obtain g x h x on L x. Let D x be the divisor in h x having maximal multiplicity at P x (being at least d x + and let W x V L x (L x 1, P x 1 be the associated 1-dimensional linear subspace. Then V (L x, P x is the inverse image of W x in V (L x 1, P x 1. One has dim(v (L x, P x dim(v (L x 1, P x 1 dim(v L x (L x 1, P x ( d x Also V (L x, P x contains the kernel of V (L x 1, P x 1 Γ(L x, O Lx (d, hence it contains l 1... l x Γ(P, O P (d x if x d. 7

8 Of course S 1 (L, P S (L, P if and only if h i g i for all 3 i e. The condition h 3 g 3 is clearly equivalent to dp 3 P(V L3 (L, P, hence it is equivalent to (L 3, P 3 V d,e. Under these equivalence we also have V (L 3, P 3 V (L 3, P 3. Let 3 < i e and assume that h j g j for 3 j < i is equivalent to (L i 1, P i 1 V d,i 1 and that under this equivalence one has V (L i 1, P i 1 V (L i 1, P i 1. Assume h j g j for 3 j i. Since V (L i 1, P i 1 V (L i 1, P i 1 we have V L i (L i 1, P i 1 V Li (L i 1, P i 1. Then the condition g i h i is equivalent to dp i P(V Li (L i 1, P i 1, which is equivalent to (L i, P i V d,i and by construction we get V (L i, P i V (L i, P i. This finishes the proof of the claim. Example 3.4. : case e 3. All components V of V d,3 (P 3 are of expected dimension 8. In order to prove this, it is enough to show that V (P 3, since dim((p 3 9 and dim V 8 by Corollary 3.3. Now since for (L, P V general, we have V L3 (L, P Γ(L 3, O L3 (d (see Proposition 1.3(b and thus ((L, P, (L 3, P 3 V d,3 for general (L 3, P 3 P, so the claim follows immediately. Example 3.5. Let d be even and consider V (L, P (P e P i L j if i j and C Γ(P, O P ( : T Pi (C L i }. It is clear that V V d,e has dimension 5 + e and thus V is of unexpected dimension if and only if e > 4. Example 3.6. Let e 3 and consider V (L, P (P e P i L j if i j and P 1,..., P e collinear}. It is clear that V is a component of V d,e of dimension e+, so it is of unexpected dimension if e > 3. In fact, in a following paper of the authors, it will be proved that this is the only component of unexpected dimension if e 4. References [1] E. Arbarello, M. Cornalba, Footnotes to a paper of Beniamino Segre, Math. Ann. 56 (1981, [] E. Ballico, Highest order flexes of nodal plane curves, JP Jour. Algebra, Number Theory & Appl. 4(3 (004, [3] G. Casnati, A. Del Centina, On certain loci of smooth degree d 4 plane curves with d-flexes, Michigan Math. J. 50(1 (00, [4] A. Tannenbaum, Families of algebraic curves with nodes, Compositio Mathematica 41(1 (1980, [5] A.M. Vermeulen, Weierstrass points of weight two on curves of genus three, Academisch Proefschrift, Universiteit van Amsterdam,