1 / 35

Brill-Noether Theory and Coherent Systems

Brill-Noether Theory and Coherent Systems. Steve Bradlow (University of Illinois at Urbana-Champaign). CIMAT, Guanajuato, Dec 11, 2006. Topics for today. Brief Introduction to Brill-Noether theory. Relation to Coherent Systems (and k-pairs). Coherent Systems moduli spaces.

marly
Download Presentation

Brill-Noether Theory and Coherent Systems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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, ….

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

  11. 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) ?

  12. (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 ]

  13. (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}

  14. __ 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 _

  15. Beyond Coherent Systems: k-pairs

  16. (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)

  17. 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

  18. 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). _

  19. (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]

  20. _ 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

  21. 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

  22. 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

  23. 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)

  24. (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]

  25. 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:

  26. __ 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 )

  27. 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)

  28. __ 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*)

  29. 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

  30. __ 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

  31. 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

  32. PETER

  33. __ END

  34. 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))

More Related