Brill-Noether Theory and Coherent Systems

1. Brill-Noether Theory and Coherent Systems Steve Bradlow (University of Illinois at Urbana-Champaign) CIMAT, Guanajuato, Dec 11, 2006

2. Topics for today • Brief Introduction to Brill-Noether theory • Relation to Coherent Systems (and k-pairs) • Coherent Systems moduli spaces • Applications to Brill-Noether theory

3. T O P I C S YSTEMS EAM F NVESTIGATING EOPLE OHERENT TOPICS not just for today Peter Newstead Vicente Munoz Vincent Mercat Oscar Garcia-Prada

4. E rank = n degree = d C genus = g ¸ ¸ = moduli space of semistable bundles M(n,d) M(n,d) Brill-Noether loci The main ingredients • C = smooth algebraic curve/Riemann surface of genus g>1 • M(n,d) = moduli space of degree d, rank n stable bundles on C Default option: (n,d) = 1 M(n,d) = M(n,d) smooth, projective, of dimension n2(g-1)+1 B(n,d,k) = { E in M(n,d) | h0(E) k } B(n,d,k) = { E in M(n,d) | h0(E) k }

5. Fundamentals If non-empty…. • Every irreducible component has dimension at least b(n,d,k) = n2(g-1)+1-k(k-d+n(g-1)) • Tangent spaces can be identified with the dual of the cokernel of the Petri map: • B(n,d,k) is smooth at E, of dimension b iff the Petri map is injective • B(n,d,k+1) lies in the singular locus of B(n,d,k)

6. Basic properties of B(1,d,k) are well understood: non-emptiness, dimension, irreducibility, smoothness M(1,d) = Jac(C) d n=1: Brill-Noether Theory for line bundles B(1,g-1,1) = Q [ACGH] • Non-emptiness of B(n,d,k) related to projective emdeddings of the curve C • Emptiness for generic curves defines subloci in moduli space of curves of genus g

7. THE BRILL-NOETHER PROJECT: Answer the basic questions in Brill-Noether theory for vector bundles on algebraic curves. • Proposed January 1, 2003 • Deadline of January 1, 2013 • The basic questions: For a general curve, n>1, k>0, and any d • Is B(n,d,k) non-empty? • Is B(n,d,k) connected, and, if not, what are its connected components? • Is B(n,d,k) irreducible, and, if not, what are its irreducible components? • What is the dimension (of each component) of B(n,d,k)? • What is the singular set of B(n,d,k)? http://www.liv.ac.uk/~newstead/bnt.html

8. 1 0 = = k h d h ¸ 2 1 1 2 0 0 ¡ ¡ ¡ g g g n n ¹ = = = = Clifford bound Riemann-Roch line 1 0 Positive expected dimension

9. (some of the) landmark contributions to date Sundaram/Laumon:k=1 1991 Teixidor I Bigas:generic curves (Teixidor parallelograms ) 1991 1995 Mukai: curves on K3 surfaces Brambila-Paz, Grzegorczyk, Newstead [BGN]:d < n 1997 1998 Bertram and Feinberg: det(E)=KC Ballico: hyperelliptic curves 1999 Mercat [M]:d < 2n 2000 Brambila-Paz, Mercat,Newstead, Ongay: extension of [BGN] and [M] results recent Teixidor, Ballico, ….

11. If a problem cannot be solved, enlarge it. Dwight D. Eisenhower 33rd President of the U.S.A.

12. 0 0 0 ( ) ( ( ) ) h k V V E H H E E ¸ ½ ½ How to enlarge the problem: Coherent Systems E in B(n,d,k) There is a rank k subspace (E,V) E A Coherent System of type (n,d,k) is a rank n, degree d bundle, E, together with a k-dimensional subspace of sections, [LePotier, Raghavendra-Vishwanath] M(n,d) ?

13. (E,V) , rank(E) = n deg (E) = d dim (V) = k ½ V H0(E) for all Stability and moduli spaces for Coherent Systems Stability for E : Stability for (E , V) : G(a,n,d,k) = Moduli space of a-stable Coherent Systems of type (n,d,k) [ GIT construction by King-Newstead ]

14. (E,V) , ¸ ½ V H0(E) (E,V) Not necessarily stable E rank(E) = n deg (E) = d dim (V) = k Relation to Brill-Noether loci G(a,n,d,k)= { a-stable coherent systems (E,V) } B(n,d,k) = { stable bundles E with h0(E) k}

15. __ d n-k • 0 < a < a1: (E,V) a -stableE semistable aL a1 a2 G0 G1 GL B(n,d,k) ( but ) Range for a (non-emptiness criterion for G(a,n,d,k) ) • At a=ai : can have (E’,V’) such that ma(E’,V’)=ma (E,V) • ai< a < ai+1 : G(a,n,d,k) independent of a a=0 _

16. Beyond Coherent Systems: k-pairs

17. (E, f1, . . . .fk) t-stable E semistable (E, f1, . . . .fk) k-pairs: stability, moduli spaces, relation to B(n,d,k) Rank (E) = n deg (E) = d fi in H0(E) • Stability for k-pairs depends on a parameter, t • Get moduli spaces K(t,n,d,k) for all t in a range • For t close to tmin: K(t,n,d,k) _ B(n,d,k)

18. Coherent Systems and k-pairs: Bundles with extra structure / Decorated bundles/ Augmented holomorphic bundles Gauge theoretic descriptions of moduli spaces: orbit spaces for a complex gauge group acting on infinite dimensional spaces of connections and bundle sections Hitchin-Kobayashi correspondence: Stability expressed by a condition involving curvature of a connection (and a contribution from bundle sections) The stability condition minimizes a (Yang-Mills-Higgs) energy functional, and corresponds to the vanishing of a Symplectic moment map

19. c YMHt : R __ How to use k-pairs to prove non-emptiness of B(n,d,k) c Gauge theory gives an (infinite dimensional) configuration space paremeterizing all holomorphic k-pairs c For fixed t can define an energy functional K(t,n,d,k) with absolute minima which satify equations corresponding to t-(poly)stability for k-pairs The hope: Given a suitable starting point, the YMHt gradient flow will terminate at a t-(poly)stable k-pair. [Daskalopoulos/Wentworth, ‘99] For small enough t, 0<k<n, k<d+(n-k)(g-1) and 0<d<n, this works and gives alternate proof of [BGN] non-emptiness results for B(n,d,k). _

20. (E,V) , rank(E) = n deg (E) = d dim (V) = k ½ __ d V H0(E) n-k aL a1 a2 G0 G1 GL B(n,d,k) • Understand difference Gj Gi for j<i The Coherent Systems way a=0 Gi = G(a,n,d,k) for ai < a < ai+1 _ Problem: G0 may be no simpler than B(n,d,k) Solution: Exploit the parameter ! • Understand Gi for some suitable i [TOPICS]

21. _ B(n,b,k) Non-emptiness of What we can gain: Irreducibility of B(n,d,k) (when non-empty) Further geometric/topological information: Pic, p1, … Framework for understanding observed features of BN theory

22. V V E E (E,V) , rank(E) = n deg (E) = d dim (V) = k O O ¸ ­ ­ ½ V H0(E) F 0 0 0 N k n : The large-a limit : Description of GL (birationally) k<n : No torsion semistable Gr(k, d+(n-k)(g-1)) GL(n,d,k) M(n-k,d) h1(F*) GL(n,d,k) = Quot scheme

23. 0 0 (E,V) (E1,V1) (E2,V2) ac- ac+ What happens at a critical value for a ac Difference between G(ac-) and G(ac+) is due to objects (E,V) which become strictly semistable at a=ac G(ac-) G(ac+) If (E,V) is ac+ -stable but not ac- -stable: • (E1,V1) and (E2,V2) are ac+ -stable but ac - semistable • Equal ac- slope • mac+(E1,V1) < mac+(E2,V2) ….with an analogous destabilizing pattern if (E,V) is ac--stable but not ac+ -stable

24. Flip loci G+(ac) = { (E,V) a-stable for a > ac but not for a < ac } ac- ac+ ac Main issue: codimension of the flip loci G(ac-) G(ac+) Flips G(ac-) - G-(ac) = G(ac+) – G+(ac) • If positive, then useful information passes between G(ac-) and G(ac+) • Combine with understanding of GL to study G0 and hence B(n,d,k)

25. (E,V) , rank(E) = n deg (E) = d dim (V) = k ½ V H0(E) G0 GL B(n,d,1) M(n-1,d) A good case (k=1 < n) Coherent systems = Vortices (stable pairs) V= Span{f} • few possible destabilizing patterns • Codimensions of flip loci can be estimated – all positive [n=2: Thaddeus] _ For 0 < d <n(g-1), B(n,d,1) is non-empty, irreducible, and of expected dimension [New proof of Sundaram]

26. d-n (E,V) , rank(E) = n deg (E) = d dim (V) = k __ Max { , 0 } n-k ½ __ d V H0(E) n-k 0 N V O ­ G(a,n,d,k) B(n,d,k) The case k < n aL aI aT a1 a=0 0 E F 0 No torsion semistable a > aL: G(a,n,d,k) is birational to a Gr(k, d+n(g-1))-bundle over M(n-k,d) a > aT : Flip loci have positive codimension a > aI : G(a,n,d,k) is smooth of dimension n2(g-1)+1-k(k-d+n(g-1)) 0 < a < a1:

27. __ d G0 GL n-k _ B(n,d,k) M(n-k,d) A good case (k<n) 0 < d < min {2n, n+ng/(k-1)} aI=aT=0 aL a1 0 aI aT • For all i, Gi birationally equivalent to GL _ • B(n,d,k) non-empty, irreducible and of expected dimension iff GL non-empty • GL non-empty iff k < d+(n-k)(g-1) ( New proof of [BGN] for d<n, and [M] for d<2n )

28. aT ac n-k (E1,V1) (E,V) (E2,V2) 0 0 no effective critical values above aT · __ d k n-2 d __ n-k The case k = n-1 has the form Any (E,V) in G (ac) + 0 < ki < ni , i=1,2 • G(a,n,d,k) = GL for all aT < a < • GL is a Gr(n-1, d+g-1)-bundle over M(1,d) = Jacd(C)

29. __ d GL Jacd(C) P pJ Jacd(C) C Jacd(C) x C The case k = n-1 n-g < d < n aT =0 n-k G0 Isomorphism outside B(n,d,n), but B(n,d,n) = f Fiber over F = Gr(n-1, H1(F*)) B(n,d,n-1) Variety of linear systems of degree d+2g-2 and dimension d+g-n-1 Gr(n-1, R1pJ*P*)

30. aT ac n-k ½ are at least g Codimensions of “flip loci” __ d - G+(ac) G(ac+) - d __ n-k For all aT < a < Other results for k < n • G(a) and GL are isomorphic outside codimension at least g • G(a) birational to GL • Pic (G(a)) = Pic (GL ) pi pi • (G(a)) = (GL ) for i < 2g - 1

31. __ d k=n-1: P(GL) =P(M(n-k,d)) . P(Gr( k, d + (n-k)(g-1) ) aT n-k for all aT < a < This gives P(G(a)) d __ n-k Yet more detailed topological information: Poincare polynomials 0 < k < n k < d+(n-k)(g-1) GL is non-empty GL is a Gr(k,d+(n-k)(g-1) – bundle over M(n-k,d) (n-k ,d) =1

32. Other CSDevelopments • Newstead and Lange: g=0 and g=1 • Brambila-Paz, Bhosle, Newstead: k=n+1 • Teixidor: n=2, k=n+2 To do: k=n, k>n

33. PETER

34. __ END

35. aT < ac < (E1,V1) (E,V) (E2,V2) 0 0 • The blow-ups along the flip loci in G(ac) G(ac-) and are the same G(ac+) G(ac-) G(ac+) d __ n-k ac- ac+ k=n-2: ac If then any (E,V) in G+ (ac) G(ac-) G(ac+) has the form : • ki = ni-1 and ac is in the torsion free range for (Ei,Vi) • For given (n1,d1), the contribution to the flip locus is a projective bundle over GL(n1,d1,n1-1)XGL (n2,d2,n2-1) This permits computation of P(G(a))