Line Bundles on Plane Curves
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- Giles Newman
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1 Stanford University Outline 1 2 3
2 The general question: Let X be a variety and L an effective line bundle. Inside of X PΓ(X, L), V L is the closed subscheme whose fiber over a line [s] PΓ(X, L) is the vanishing locus of s. Study Pic V L /PΓ(X,L) that is, line bundles supported on divisors D in the linear system L. In particular, what is the geometry of pairs (D cl X, L ), where D L and L is a line bundle on D, say with h 0 (D, L ) > k? Preliminaries For today, X = P 2 L = O(d) C d := V L PΓ(X, L) is the universal plane curve of degree d Studying geometry of a cohomological stratification of Pic Cd /PΓ(X,L) Warning! Pic Cd /PΓ(X,L) is not representable by a scheme!
3 () Why study this? Birational geometry of Pic Cd /PΓ(X,L) is important in algebra its function field is the center of the generic division algebra. (Procesi) We ll see that these spaces are unirational. Rationality is open in general. Cohomological loci in Pic V L /PΓ(X,L) give information on high-codimension cycles in X, as we ll see later. Intrinsically interesting! () How do we study these questions? Parametrize minimal free resolutions (essentially due to Beauville, Determinental Hypersurfaces) Relate cohomological loci to Hilbert schemes of points
4 Outline Suppose ι : D P 2 is a plane curve, and L is a line bundle on D. Strategy: Parametrize minimal free resolutions of k[x 0, x 1, x 2 ]-modules of the form H 0 (P 2, ι L(n)) := n Z Γ(P 2, ι L(n)). Definition (Minimal Free Resolution) Let M be a (graded) module over k[x 0, x 1,..., x n ]. Recall: a f.g. (graded) free resolution K q.i. M is minimal if all the maps in K are zero mod the irrelevant ideal (x 0, x 1,..., x n ). Canonical!
5 (cont.) Suppose 0 E i O(d i ) e i ι L 0 is exact. Note: E is a vector bundle (can be checked locally). Can we arrange it that E splits into a direct sum of line bundles? (cont.) 0 E i O(d i ) e i ι L 0 For convenience, if F is a quasi-coherent sheaf on P n, define H i (F) := k Z H i (P n, F(k)). Theorem (Horrocks) Suppose E is a vector bundle on P n. Then E splits into a direct sum of line bundles if and only if for 1 i n 1. H i (E) = 0
6 (cont.) 0 E i O(d i ) e i ι L 0 Want: H 1 (E) = 0. From the LES in cohomology, this happens iff H 0 ( i O(d i ) e i ) H 0 (ι L) is surjective. Can arrange this simply by picking (homogeneous) generators for H 0 (ι L), as the images of the generators of H 0 (O(d i )). Dictionary 0 i O(d i,2 ) e i,2 f i O(d i,1 ) e i,1 ι L 0 ι L is supported on C := V (det(f )), ( ( )) det(f ) Γ P 2, O d i,1 e i,1 d i,2 e i,2 i χ(c, L) = χ(ι L) = i (e i,1 χ(o(d i,1 )) e i,2 χ(o(d i,2 ))) so can determine degree of L by Riemann-Roch h 0 (C, L) = i (e i,1 h 0 (O(d i,1 )) e i,2 h 0 (O(d i,2 ))) If deg C (L) < deg(c), L is generated by global sections iff d i,1 = 0 for all i.
7 Geometric Implications Corollary Let P d h,k (T ) = {L Picd C k /PΓ(P 2,O(k)) (T ) rk T (L) h}, G j = i GL(e i,j ) For an appropriate choice of d i,j s, there is a rational, birational map G 2 \ Hom(O(d j,2 ) ej,2, O(d i,1 ) ei,1 )/G 1 Ph,k d. i,j d i,1 >d j,2 Corollary P d h,k is unirational for all d, h, k, etc. Outline 1 2 3
8 Hilbert Scheme of Points Suppose S/k is a smooth projective surface. Definition (Hilbert Scheme of Points) Hilb n S(T ) = {Closed subschemes of S T, flat over T, with fibers of length n}. Hilb n S is smooth, connected, and projective geometry key to understanding the 0-cycles on S. The interesting part of the geometry of Hilb n S is contained in the geometry of line bundles on curves in S. Hilbert Scheme of Points Suppose L is an effective line bundle on S, and consider the map r : V L PΓ(S, L) as before. Natural map p : Hilb n PΓ(S,L) V L Hilb n S If I n is the ideal sheaf of the universal family F n S Hilb n S, and L is extremely ample, this is the same as P(π 2 (π 1L I n )) Hilb n S. But Hilb n PΓ(S,L) V L has a natural stratification over Ph,k L locus of f PΓ(S, L) with V (f ) smooth). (over the
9 Hilbert Scheme of Points Restrict to r : V L sm Natural map r : V L PΓ(S, L) PΓ(S, L) sm where r is smooth. Hilb n PΓ(S,L) sm V sm L Pic n PΓ(S,L) sm V sm L, which is a P h 1 (Zariski!)-fiber-bundle over Ph,n L,sm \ P L,sm line bundles L supported on smooth curves C L with h 0 (L ) = h). h+1,n (that is, Grothendieck Ring of Varieties Warning! This will only work in characteristic zero. Let k be a field, and let K 0 (Var k ) be the free abelian group on isomorphism classes of k-varieties, subject to the relation: [X ] = [Y ] + [X \ Y ] if Y is a closed subvariety of X. This is a ring via [X ] [Y ] := [X Y ]. Motivic Zeta Function (Kapranov): Z mot X (t) K 0(Var k ) ZX mot (t) := n [Sym n (X )]t n
10 Göttsche s Theorem ZX mot (t) := n [Sym n (X )]t n Example (The Projective Plane) Z mot P 2 (t) = Theorem (Göttsche) 1 (1 t)(1 [A 1 ]t)(1 [A 2 ]t) Let S be a smooth projective surface/k a field of characteristic zero. Then [Hilb n (S)]t n = n m=1 Z mot S ([A m 1 ]t m ). Generating Functions From before, we have d [Hilb d PΓ(P 2,O(n)) sm C sm n ]t d = d ( h [P h 1 ][P d,sm h,n ) \ Ph+1,n d,sm ] t d Theorem (R. Vakil, M. Wood) Let P X be a closed subscheme of finite length, L an ample line bundle. Let Γ(X, L) sm P be the constructible subset consisting of sections vanishing on P whose zero locus is smooth. Then lim d Γ(X, L d ) sm P Γ(X, L d ) tends to an (explicit) limit in a certain completion K0 (Var k )[A 1 ] 1.
11 Generating Functions Putting this together gives a certain explicit sequence of elements S d K0 (Var k )[A 1 ] 1 (Laurent series in [A 1 ] 1 )so that ( ) lim S d [P h 1 ][P n h,n d,sm \ Ph+1,n d,sm ] t d = d h S d [Hilb d PΓ(P 2,O(n)) C sm sm n ]t d mod codim 3 = m=1 lim n d d [Hilb d (P 2 )] [A 2d t d mod codim 3 = ] 1 (1 [A 1 ] m 1 t m )(1 [A 1 ] m t m )(1 [A 1 ] m+1 t m )
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