Algebraic Geometry (Math 6130)


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1 Algebraic Geometry (Math 6130) Utah/Fall Projective Varieties. Classically, projective space was obtained by adding points at infinity to n. Here we start with projective space and remove a hyperplane, leaving us with an affine space. Even though they are not functions on projective space, there is a welldefined notion of vanishing of a homogeneous polynomial at a point of projective space. This allows us to define algebraic sets and varieties in projective space analogous to the algebraic sets and varieties in 1. Let V be a finitedimensional vector space over a field and denote the dual space Hom(V, ) by V. In the 1960s and 70s, there was some transatlantic controversy over whether the projective space associated to V ought be the set of minimal subspaces (lines through the origin) in V or else the set of minimal quotient spaces of V. We ll use both definitions here (with different fonts to distinguish them). Definition 2.1. (i) (American projective space) Let P(V ) be the set of lines through the origin in V, i.e. P(V ) = (V {0})/ where v v if if v = λv for some λ. (ii) (European projective space) Let P(V ) be the set of onedimensional quotients of V, i.e. P(V ) = (V {0})/ where (V q ) (V q ) if there is a linear isomorphism (multiplication by λ ) such that q = λ q. λ Remars. (a) To each nonzero (hence surjective) map V q, we may associate the ernel W = er(q) V, which is a maximal subspace, i.e. a hyperplane in V. Conversely, to each hyperplane W V, the quotient V V/W = is uniquely determined up to the equivalence. So P(V ) may be thought of as the set of hyperplanes in V. (b) Clearly there is a natural identification P(V ) = P(V ), but just as a vector space is not canonically isomorphic to its dual, there is no canonical isomorphism between P(V ) and P(V ). On the other hand, once an isomorphism V = V is chosen (e.g. from a choice of basis) then an isomorphism between P(V ) and P(V ) results. This bit of pedantry will be important when we replace vector spaces with vector bundles (and the reader may have noticed that we have not yet even defined an isomorphism of projective spaces). 1
2 2 Example 2.1. (a) P() = P() is a point. (b) A short exact sequence of vector spaces: ( ) 0 U f V g W (= V/U) 0 determines inclusions: P(U) P(V ) and P(W ) P(V ). These are the projective subspaces of P (V ) and P(V ) respectively. Maximal projective subspaces are called projective hyperplanes. In particular, if P (U) P (V ) is a hyperplane then W is a line, and in that case we lose no generality by letting W =. (c) Suppose W = in ( ). A map l : V splits the sequence ( ) if the composition l V g is nonzero. A pair of splittings l, l differ (after a uniquely determined scaling of l ) by an element l l : U. This gives the set of splittings the structure of an affine space with U as the underlying vector space. In a bit more detail, a splitting l gives rise to an isomorphism: (f, l) : U V The points of P (V ) P (U) may be identified with maps l : U (using the isomorphism above) with l (1) = (u, 1). In other words, once the splitting l is chosen, we have: P (V ) P (U) = U with the chosen splitting l representing the origin in U. Without the choice of splitting, P (V ) P (U) is only affine space. This is important enough to record it as: Proposition 2.1: Each point of P(V ) yields a sequence: ( ) 0 U V 0 and a projective hyperplane P (U) P (V ). The complement of P (U) is an affine space with underlying vector space U, which acquires an origin with a choice of p P (V ) P (U) (splitting the sequence ( )). Conversely, given a vector space U, we may let V = U and then: P (U) U = P (V ) is a disjoint union of U and a projective space of lines in U. Definition 2.2. Projective nspace is P n = P (n+1 ). It contains: coordinate hyperplanes H i = P (U i ) and their complements U i P n for each i = 0,..., n corresponding to the split sequences: 0 U i = e j j i n+1 e i 0
3 Remars. (a) Projective nspace is covered by subsets that we may identify with the vector spaces U i : n P n = with [l : n+1 ] U i if and only if l(1) = n j=0 a je j with a i 0. (b) Projective nspace is a disjoint union of vector spaces: P n = U 0 P n 1 i=0 U i = U 0 (U 01 P n 2 ) = = U 0 U 01 U 0...n where U 01...,i = e i+1,..., e n n+1 and in particular U 0...n = {0}. Warning! Do not confuse the subspaces U 0...i = U 0 U i n+1 with the subsets U 0 U i P n. The former are vector spaces of dimension n i, and the latter (as we will see) are all open, dense subsets of P n of the ssame dimension n Notation 2.1. If l : n+1 with l(1) = n i=0 a ie i 0, then: (a 0 : : a n ) P n are the projective coordinates of the point associated to l, with the colons between coordinates indicating the ambiguity: (a 0 : : a n ) = (λa 0 : : λa n ) for λ Next we revisit homogeneous polynomials in this context. Observation. If F S d = [x 0,..., x n ] d and 0 a n+1, then F (a) 0 F (λa) = λ d F (a) = 0 for all λ, hence V (F ) n+1 is a union of lines through the origin and more generally, V (I) n+1 is a union of lines through the origin if I = F 1, F m S is a homogeneous (graded) ideal. Conversely, Definition 2.3. A subset C V of a vector space is a cone over the origin if C is a union of lines through the origin (the rulings of C). Proposition 2.2. The ideal I(C) S of a cone over the origin in n+1 is homogeneous if is an infinite field. Proof. For f S, let f = f f d be the decomposition of f into homogeneous terms. Then: f(λa) = f 0 (a) + λf 1 (a) + + λ d f d (a) Thus if f(a) = 0 for a 0, then f(λa) vanishes for all (infinitely many) values of λ, i.e. it has infinitely many roots, as a polynomial in λ, so. it is the zero polynomial. It follows that if f I(C), then each f e I(C), i.e. I(C) is a homogeneous ideal. 3
4 4 Definition 2.4. Given a homogeneous ideal I S, then: V h (I) = {rulings of the cone V (I) n+1 } P n is the associated algebraic set, and given X P n, let C(X) n+1 be the cone over the origin with rulings indexed by X, and: I(X) := I(C(X)) S is the associated geometric homogeneous ideal (assuming is infinite). Convention. If X =, let C(X) = {0} and I(X) = m = x 0,..., x n. This is often called the irrelevant homogeneous ideal. It is the only homogeneous ideal that is also a maximal ideal in the traditional sense. Let be algebraically closed (hence in particular, infinite). Nullstellensatz for P n. Homogeneous ideals I x 0,..., x n satisfy: I(V h (I)) = I Proof. This follows from the Nullstellensatz (or rather Corollary 1.2) whenever V h (I), and from the convention when V h (I) =, since we require that I x 0,..., x n so that in that case, V (I) = {0}. Corollary 2.1. V h (I) = if and only if there is an N > 0 such that x N i I for all i = 0,.., n. Via the Nullstellensatz, we have graded correspondences: X I(X) [X] := S/I(X) among subsets of P n, homegeneous ideals in S and quotient rings of S: algebraic sets radical homogeneous ideals reduced graded quotients projective varieties homogeneous primes graded quotient domains points submaximal homogeneous ideals of the form I = l 1,.., l n where l 1,.., l n S 1 are n linearly independent forms. The graded quotient ring S R corresponding to a point has Hilbert function dim(r d ) = 1 for all d 0 and graded Koszul resolution: ( n 2) n 0 S( n) S( 2) S( 1) S R 0 Definition 2.5. (a) The graded homogeneous coordinate ring of a projective variety X P n is the graded domain: [X] := [x 0,..., x n ]/I(X) which coincides with the coordinate ring of the cone C(X) n+1.
5 (b) The field of rational functions (X) of X P n { } is: F (X) = G F, G [X] d for some d, G 0 (C(X)) Proposition 2.2. The fields (X) and (C(X)) of rational functions associated to a (nonempty!) projective variety X = V h (P ) P n and the cone C(X) = V (P ) n+1 over the origin are related by: (C(X)) = (X)(x) Proof. Define the integergraded ring: { } F R = G F [X] d, G [X] e, G 0 (C(X)) and choose 0 x [X] 1 which is always possible if X. Then: (i) R is isomorphic to (X)[x, x 1 ] (graded by the power of x) (ii) The ordinary field of fractions of R is (C(X)). from which the Proposition immediately follows. Definition 2.6. Let X P n be a projective variety. Then: (a) dim(x) is the transcendence degree of (X) over. (b) X is nonsingular at a X if: (( ) ) Fi ran (a) = n dim(x) x j i.e. the ran of the Jacobian matrix agrees with the codimension of X in P n, where P = I(X) = F 1,..., F m (with homogeneous generators). The variety X is nonsingular if it is nonsingular at every point. Remars. (i) The field (a) = when a P n is a point! (ii) By Proposition 2.2, the dimension of X satisfies: dim(x) + 1 = dim(c(x)) where C(X) is the associated cone in n+1. (iii) The point a X is nonsingular if and only if every nonzero point a C(X) in the ruling corresponding to a is nonsingular since the codimension of X in P n agrees with the codimension of C(X) in n+1 and the Jacobian matrices are the same! (iv) In the next section, we will cover projective varieties X with affine varieties X U i, and see that there, too, the definitions above are sensible if we thin of the X U i as being local affine charts of the global projective variety. 5
6 6 Definition 2.7. A nonsingular projective variety X is a geometric model for its function field (X). Theorem (Resolution of Singularities): If is algebraically closed of characteristic zero, then every finitely generated field K of finite transcendence degree over has a nonsingular geometric model. Remar. This is unnown when char() = p and tr deg(k/) 4. Birational algebraic geometry extracts information about a field K from its nonsingular geometric models. We end with some basic examples from (multi)linear algebra. Example 2.1 (Continued) A projective subspace P (U) P n coming from a sequence of vector spaces: 0 U n+1 W 0 is a projective variety, cut out by the linear forms in W ( n+1 ). If we let y 0,..., y m be a basis for U = ( n+1 ) /W, then: (P (U)) = (y 1,..., y m ) and the cone over the origin in n+1 with rulings indexed by P (U) is, simply, U n+1 itself, with (U) = (y 0,..., x m ). Example 2.2. Quadrics. Consider a quadratic polynomial: F = i>j c ij x i x j S 2 = [x 0,..., x n ] 2 We may convert F into a symmetric matrix: Γ F := (γ ij ) with γ ii = c ii and γ ij = 1 2 c ij for i j With this conversion, we see that: F (a 0 :... : a n ) = 0 (a) T Γ F (a) = 0 i.e. the points of the quadric hypersurface Q = V (F ) P n are the lines λa n+1 that are isotropic with respect to the inner product defined by Γ F. In particular, if U = er(γ F ) n+1, then P (U) Q. In fact, P (U) Q is the locus of singular points of the quadric Q. In particular, when Γ F is invertible, tthe quadric Q is nonsingular. Since we assume that is algebraically closed, the GrammSchmidt process for the inner product yields a basis for n+1 with respect to which the inner product diagonalizes with 1 s and 0 s on the diagonal. That is, with respect to this new basis v 0,..., v n n+1 and dual basis w 0,..., w n ( n+1 ), F = w w 2 m and the singular locus of Q is the projective subspace P ( e m+1,..., e n ).
7 7 We may also consider the projective space of quadrics. This is: P (n+2 2 ) 1 for (n+2 2 ) = Sym 2 ( n+1 ) with basis e i e i and e i e j + e j e i with dual basis (x ij ; i < j), so this projective space is associated to the ring: S = [x ij ; i < j] Via the Nullstellensatz, specifying a quadric Q P n is the same as specifying its homogeneous ideal c ij x i x j up to scalar multiple, so the quadrics are in bijection with he points P (n+2 2 ) 1 via: Q (γ ij ) P (n+2 2 ) 1 with the exception of the polynomials c ij x i x j that are perfect squares. But these are the image of the Veronese embeddings: v n,2 : P n P (n+2 2 ) 1 ; a i e i ( a i e i ) 2 This image is a projective variety that is an intersection of quadrics. Subexamples. (i) In the case n = 1, we have: v 1,2 : P 1 P 2 ; (a 0 : a 1 ) (a 2 0 : a 0 a 1 : a 2 1) and the image is the conic C = V (x 00 x 11 x 2 01) P 2. (ii) In the case n = 2, we have the Veronese surface: v 2,2 : P 2 P 5 ; (a 0 : a 1 : a 2 ) (a 2 0 : a 0 a 1 : a 2 1 : a 0 a 2 : a 1 a 2 : a 2 2) and the equations for the image are the quadrics appearing as the 2 2 minors of the following symmetric matrix: A = x 00 x 01 x 02 x 01 x 11 x 12 x 02 x 12 x 22 Here are a couple of interesting features of this example: (i) There are five independent quadratic polynomials among the 2 2 minors of A, and these generate the homogeneous ideal I(v 2,2 (P 2 )), yet v 2,2 (P 2 ) P5 has codimension three. In general, a homogeneous ideal P S requires more generators than the codimension of the variety X = V (P ) P n. The rare homogeneous ideals P that are generated by codim(x) generators are called complete intersection ideals. (ii) The cubic hypersurface in P 5 cut out by G = det(a) parametrizes the locus of singular quadrics in P 2, together with the image of v 2,2. This is itself a singular hypersurface, singular along the image of v 2,2!
8 8 Example 2.3 duple embeddings generalize the Veronese embeddings. Let V = n+1, and consider the mapping: v n,d : P (V ) P (Sym d V ); v n,d ( a i e i ) = ( a i e i ) d to the space of symmetric tensors. With the natural basis for: we have: v n,d ( : a i : ) = ( : Sym d (V ) V d n i=0 a d i i : ) for d d n = d i.e. it is given by the collection of a monomials of degree d in the a i. From this, it is not difficult to see that the image of v n,d is always cut out by quadratic polynomials, and indeed the homogeneous ideal I(v n,d (P n )) is generated by quadratic polynomials. The case n = 1 is particularly nice. Here: v 1,d (a 0 : a 1 ) = (a d 0 : a d 1 0 a 1 : : a 0 a d 1 1 : a d 1) and the image is the ran one locus of the matrix: [ x0 x 1 x n 1 The image of v 1,d in P d x 1 x 2 x n ] is the rational normal curve of degree d. Example 2.4 Segre embeddings are of the form: s V,W : P (V ) P (W ) P (V W ); (λa, µb) λµa b In coordinates, if V = m+1 and W = n+1, then: s m,n ((a 0 :... : a m ), (b 0 : b n )) = ( a i b j ) and once again the image of s m,n is cut out by quadratic polynomials and the homogeneous ideal is generated by quadratic polynomials. For example, the image of s 1,1 : P 1 P1 P3 is: { } (a0 b 0 : a 1 b 0 : a 0 b 1 : a 1 b 1 ) P 3 = V (x0 x 3 x 1 x 2 ) which is the nonsingular quadric in P 3. Segre embeddings can be generalized to finite sets of vector spaces: s V1,...,V m : P (V 1 ) P (V m ) P (V 1 P (V m ) in the obvious way.
9 Example 2.5. The Grassmannian Let V be a vector space over of dimension n. Then G(m, V ) is the subset: G(m, n) P ( m V ) of decomposable elements. That is, points of G(m, n) are of the form: ( ) ( 1) sgn(σ) v σ(1) v σ(m) ; v 1,..., v m V σ Σ m for linearly independent vectors v 1,..., v m V. The points of the Grassmannian are in a natural bijection with the set of subspaces of V, so that for example: G(m, V ) = P (V ) and G(n 1, V ) = P(V ) Once again, the Grassmannian is cut out by (Plücer) quadrics both settheoretically and idealtheoretically, and the Grassmannians G(m, V ) are projective varieties. This involves some combinatorics, but the case m = 2 and dim(v ) = 4 is particularly simple. Here, G(2, 4) P ( 2 V ) is a subset of the space of sewsymmetric maps, and: 0 γ 12 γ 13 γ 14 G(2, 4) P γ 12 0 γ 23 γ 24 γ 13 γ 23 0 γ 34 γ 14 γ 24 γ 34 0 is the quadric cut out by the Pfaffian of the matrix of indeterminates: x 12 x 34 x 13 x 24 + x 14 x 23 The Grassmannian G(m, V ) is the image of the mapping V m U P ( m V ) defined by ( ) above. This is welldefined on the subset: U V m of linearly independent mtuples of vectors, which is the complement of an algebraic subset of V m = ( n ) m. 9
10 10 Exercises Consider the inclusion: n = e 1,..., e n P n 0 U 0 n+1 0 associated to: and the splitting e 0 : n+1. This geometry is associated to the process of homogenizing and dehomogenizing polynomials: Deomogenizing. The dehomogenization of F S d is: f(x 1,..., x n ) := F (1, x 1,..., x 0 ) Homogenizing. For each f [x 1,..., x n ] and d deg(f), ( F (x 0,..., x n ) := x d x1 0f,, x ) n S d x 0 x 0 produces a homogeneous polynomial that dehomogenizes to f. (a) Show that dehomogenizing and homogenizing all the elements of an ideal converts homogeneous ideals in S to homogeneous ideals in [x 1,..., x n ] and vice versa. Notice that on the level of ideals, there is no ambiguity...all values of d deg(f) must be chosen when homogenizing an ideal containing f. (b) Show that radical ideals and prime ideals are preserved under homogenizing and dehomogenizing. Moreover, show that: I [x 1,..., x n ] hom(i) S dehom(hom(i)) [x 1,..., x n ] returns the ideal I. What happens to a homogeneous ideal under: I S dehom(i) [x 1,..., x n ] hom(dehom(i) S 2.2. (a) Under the bijections between ideals and varieties, we see from Exercise 2.1 (b) that the set of algebraic subsets of n is in bijection (via homogenization) with a subset of the set of algebraic subsets of P n. What geometric property must an algebraic subset of Pn have in order to be in the image of this identification? (b) Consider the twisted cubic curve in 3 : C = {(s, s 2, s 3 ) s } 3 Identify the curve C P 3 associated to C via homogenizing, and show that I(C) requires only two generators, whereas I(C) = hom(i) requires three generators. Thus, it is not sufficient to homogenize the generators of an ideal to find the generators of the homogenized ideal.
11 2.3. (a) If X n is a variety and X P n is the associated projective variety, find an isomorphism between the field of fractions: (X) = (X) (b) Show that a point x X is nonsingular as a point of X if and only if it is nonsingular when viewed as a point x X of X. Hint: Try it first for a hypersurface. Euler s identity may be useful: n F df = x i for F S d x i i=0 (c) Consider the nonsingular curve: d C = V (y 2 (x λ i ) for distinct roots λ 1,..., λ d. Find the complement C C P 2 and decide whether the points of the complement are singular or not. i=1 (d) Find C for C = V (y x 3 ) and show that it has a singular point Show that the singular locus of the quadric Q = V (F ) P n for F S 2 {0} is precisely the projective subspace P (er(γ F )), where Γ F is the associated symmetric matrix Show that the images of the duple and Segre embeddings are nonsingular projective varieties. Show that each G(m, V ) is a variety. Show that the ideals of each of these is generated by quadratic polynomials Let G be the determinant of the matrix: A = x 00 x 01 x 02 x 01 x 11 x 12 x 02 x 12 x 22 Show that the singular locus of X = V (G) is the Veronese surface. 11
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