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

• Applications to Brill-Noether theory

EAM

NVESTIGATING

OHERENT

YSTEMS

EOPLE

F

T O P I C S

Oscar Garcia-Prada

Peter Newstead

Vicente Munoz

Vincent Mercat

TOPICS not just for today

The main ingredients

• C = smooth algebraic curve/Riemann surface of genus g>1

M(n,d) = moduli space of semistable bundles

• M(n,d) = moduli space of degree d, rank n stable bundles on C

(n,d) = 1 M(n,d) = M(n,d)

M(n,d) smooth, projective, of dimension n2(g-1)+1

Default option:

B(n,d,k) = { E in M(n,d) | h0(E) k }¸

B(n,d,k) = { E in M(n,d) | h0(E) k }¸

rank = n

degree = d

genus = g

E

C

Brill-Noether loci

Fundamentals

• Every irreducible component has dimension at least

(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 iff the Petri map is injective

• B(n,d,k+1) lies in the singular locus of B(n,d,k)

If non-empty….

n=1: Brill-Noether Theory for line bundles

M(1,d) = Jac(C)d

B(1,g-1,1) =

Basic properties of B(1,d,k) are well understood: non-emptiness, dimension, irreducibility, smoothness [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

THE BRILL-NOETHER PROJECT:

Answer the basic questions in Brill-Noether theory for

vector bundles on algebraic curves.

http://www.liv.ac.uk/~newstead/bnt.html

• 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)?

Clifford bound

1

g¡ 1

Riem

ann-R

och li

neg¡ 1

h0=0

k=n=¸

d=n=¹0

h1=0

2g¡ 2

Positive expected dim

ension

(some of the) landmark contributions to date

Brambila-Paz, Grzegorczyk, Newstead [BGN]: d < n

Brambila-Paz, Mercat, Newstead, Ongay: extension of [BGN] and [M] results

1997

Teixidor I Bigas: generic curves (Teixidor parallelograms )1991

Mercat [M]: d < 2n

2000

1999

1995 Mukai: curves on K3 surfaces

1998 Bertram and Feinberg: det(E)=KC

recent Teixidor, Ballico, ….

Sundaram/Laumon: k=11991

Ballico: hyperelliptic curves

If a problem cannot be solved, enlarge it.

Dwight D. Eisenhower 33rd President of the U.S.A.

There is a rank k subspace V ½H0(E)

h0(E) ¸ k

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]

How to enlarge the problem: Coherent Systems

?M(n,d)

E in B(n,d,k)

V ½H0(E)

E (E,V)

Stability and moduli spaces for Coherent Systems

for all

Stability for E :

Stability for (E , V) :

G(,n,d,k) = Moduli space of stable Coherent Systems of type (n,d,k)

rank(E) = n

deg (E) = d

dim (V) = k

½V H0(E)(E,V) ,

[ GIT construction by King-Newstead ]

Relation to Brill-Noether loci

G(,n,d,k)= { -stable coherent systems (E,V) }

rank(E) = n

deg (E) = d

dim (V) = k

½V H0(E)(E,V) ,

B(n,d,k) = { stable bundles E with h0(E) k}¸

(E,V)

E

Not necessarily stable

Range for (non-emptiness criterion for G(,n,d,k) )

=0

G0 GLG1

L 1 2

• At : can have (E’,V’) such that (E’,V’)= (E,V)

• i i+1G(,n,d,k) independent of

• : (E,V) stableE semistable

B(n,d,k)

n-kd__

( but )

_

Beyond Coherent Systems: k-pairs

• Stability for k-pairs depends on a parameter,

• Get moduli spaces K(,n,d,k) for all in a range

• For close to min:

(E,1, . . . .k) -stable E semistable

K(,n,d,k)

B(n,d,k)

k-pairs: stability, moduli spaces, relation to B(n,d,k)

(E, 1, . . . .k)

i in H0(E)

Rank (E) = n

deg (E) = d

_

Gauge theoretic descriptions of moduli spaces: orbit spaces for a complex gauge group acting on infinite dimensional spaces of connections and bundle sections

Coherent Systems and k-pairs:

Hitchin-Kobayashi correspondence: Stability expressed by a condition involving curvature of a connection (and a contribution from bundle sections)

Bundles with extra structure / Decorated bundles/ Augmented holomorphic bundles

The stability condition minimizes a (Yang-Mills-Higgs) energy functional, and corresponds to the vanishing of a

Symplectic moment map

How to use k-pairs to prove non-emptiness of B(n,d,k)

K(,n,d,k)

For fixed can define an energy functional

YMH: R

with absolute minima which satify equations corresponding to -(poly)stability for k-pairs

The hope: Given a suitable starting point, the YMH gradient flow will terminate at a (poly)stable k-pair.

[Daskalopoulos/Wentworth, ‘99] For small enough , 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).

__

Gauge theory gives an (infinite dimensional) configuration space paremeterizing all holomorphic k-pairs

_

The Coherent Systems way

=0

G0 GLG1

L 1 2

B(n,d,k)

n-kd__

Gi = G(,n,d,k) for i < < i+1

rank(E) = n

deg (E) = d

dim (V) = k

½V H0(E)(E,V) ,

Problem: G0 may be no simpler than B(n,d,k)

Solution: Exploit the parameter !

[TOPICS]

• Understand Gi for some suitable i

• Understand difference Gj Gi for j<i

_

What we can gain:

Irreducibility of B(n,d,k) (when non-empty)

Further geometric/topological information: Pic, 1, …

Framework for understanding observed features of BN theory

Non-emptiness of B(n,b,k)_

The large- limit : Description of GL (birationally)

GL(n,d,k)

M(n-k,d)

Gr(k, d+(n-k)(g-1))

0 F 0V E­ O k<n :

N0 k n :¸

V E­ O

GL(n,d,k) = Quot scheme

rank(E) = n

deg (E) = d

dim (V) = k

½V H0(E)(E,V) ,

h1(F*)

No torsionsemistable

Difference between G(c) and G(c

) is due to objects (E,V) which become strictly semistable at c

What happens at a critical value for c

G(c)G(c

)c

c

00 (E1,V1) (E,V) (E2,V2)

• Equalc- slope

• (E1,V1) and (E2,V2) are c-stable but c - semistable

If (E,V) is c-stable but not c

-stable:

• c+(E1,V1) < c+(E2,V2)

….with an analogous destabilizing pattern if (E,V) is c

-stable but not c-stable

G+(c) = { (E,V) -stable for c but not for c }

G(c) - G(c) = G(c

) – G(c)

c

G(c)G(c

)c

c

Main issue: codimension of the flip loci

• If positive, then useful information passes between G(c

) and G(c)

Flips

Flip loci

• Combine with understanding of GL to study G0 and hence B(n,d,k)

A good case (k=1 < n)

• few possible destabilizing patterns

• Codimensions of flip loci can be estimated – all positive

V= Span{} Coherent systems = Vortices (stable pairs)

G0 GL

B(n,d,1) M(n-1,d)

For 0 < d <n(g-1), B(n,d,1) is non-empty, irreducible, and of expected dimension

[New proof of Sundaram]

rank(E) = n

deg (E) = d

dim (V) = k

½V H0(E)(E,V) ,

[n=2: Thaddeus]

_

The case k < n

=0 L n-kd__

0 F 0No torsion

V ­ O E

semistable

TI

0 N

G(,n,d,k) is smooth of dimension n2(g-1)+1-k(k-d+n(g-1))

Flip loci have positive codimension

G(,n,d,k) is birational to a Gr(k, d+n(g-1))-bundle over M(n-k,d)

I :

L:

T :

1: G(,n,d,k) B(n,d,k)

1

n-k d-n__Max { , 0 }

rank(E) = n

deg (E) = d

dim (V) = k

½V H0(E)(E,V) ,

I

A good case (k<n)

I=T=0

0 L n-kd__

T

1

• GL non-empty iff k < d+(n-k)(g-1)

0 < d < min {2n, n+ng/(k-1)}

• For all i, Gi birationally equivalent to GL

( New proof of [BGN] for d<n, and [M] for d<2n )

• B(n,d,k) non-empty, irreducible and of expected dimension iff GL non-empty

G0 GL

B(n,d,k) M(n-k,d)_

_

The case k = n-1

no effective critical values above T

Any (E,V) in G (c) has the form

0 < ki < ni , i=1,2

• G(,n,d,k) = GL for all T < < n-k

d__

T n-k

d__c

k n-2 ·

(E1,V1) (E,V) (E2,V2)0 0

• GL is a Gr(n-1, d+g-1)-bundle over M(1,d) = Jacd(C)

The case k = n-1

n-g < d < n

T n-k

d__=0

G0 GL

Jacd(C)

Fiber over F = Gr(n-1, H1(F*))

Isomorphism outside B(n,d,n), but B(n,d,n) =

B(n,d,n-1)

Gr(n-1, R1pJ*P*) P

Jacd(C) x C CpJ

Variety of linear systems of degree d+2g-2 and dimension d+g-n-1

Jacd(C)

• (G( = (GL )

Other results for k < n

i

T n-k

d__c

-G(c) G(c

)-½Codimensions of “flip loci” are at least g

• G( and GL are isomorphic outside codimension at least g

For all T < < n-k

d__

• G( birational to GL

• Pic (G( = Pic (GL )

i for i < 2g - 1

0 < k < n

Yet more detailed topological information: Poincare polynomials

P(GL) =P(M(n-k,d)) . P(Gr( k, d + (n-k)(g-1) )

This gives P(G()) for all T < < n-kd__

k=n-1:

n-kT

d__

GL is non-empty k < d+(n-k)(g-1)

(n-k ,d) =1 GL is a Gr(k,d+(n-k)(g-1) – bundle over M(n-k,d)

Other CS Developments

• 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

PETER

END

__

k=n-2:

This permits computation of P(G())

T < c < n-kd__ then any (E,V) in G (c)

has the form :

• ki = ni-1 and c is in the torsion free range for (Ei,Vi)

If

0 0(E1,V1) (E,V) (E2,V2)

• 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)

c

G(c)G(c

)c

c

G(c)

• The blow-ups along the flip loci in

and G(c) are the same

G(c) G(c

)

G(c)