# Syzygies in Algebraic Geometry, with an exploration of a connection with String Theory (12w5117)

Arriving in Banff, Alberta Sunday, August 12 and departing Friday August 17, 2012

## Organizers

Lawrence Ein (University of Illinois at Chigago)

David Eisenbud (Mathematical Sciences Research Institute)

Gavril Farkas (Humboldt Universität zu Berlin)

Irena Peeva (Cornell University)

## Objectives

The first focus of the workshop is based on the amazing convergence of interest in Cohen-Macaulay modules (especially matrix factorizations and their generalizations) between the String Theory community and the Commutative Algebra/Algebraic Geometry community. The relatedness of these areas is not easy to penetrate, so we plan to have a short course, probably one long lecture each day of the meeting, exposing this subject from multiple points of view. The workshop could be a defining moment in this subject.

A major recent development in free resolutions came from a group of conjectures of Mats Boij and Jonas Soederberg. The original motivation of Boij and Soederberg was an attempt (now completely accomplished by these methods) to prove the Multiplicity Conjecture of Herzog, Huneke, and Srinivasan. That conjecture required some understanding of the shape of free resoluitions of graded modules over polynomial rings. The key insight of Boij and Soederberg was that it might be easier to look at the convex cone generated by the Betti tables of such resolutions than at the resolutions themselves. The original conjectures were proven by Eisenbud and Schreyer. Many other people have contributed to a very rapid development over the last several years. The current result is that we know the cone of Betti tables precisely, and also the cone of cohomology tables of vector bundles on projective spaces. We also have an understanding, only slightly less precise, of the cone of cohomology tables of coherent sheaves. The new understanding is quite orthogonal to the ways in which people had looked at these before. The area is still developing fast. Work has begun in the direction of infinite resolutions, and on other graded rings; for example, Eisenbud and Schreyer have worked on the case of cohomology tables of vector bundles on the next simplest toric variety, $P^1 x P^1$. The Boij-Soederberg theory will be another focus of our workshop.

The systematic use of syzygy techniques in order to uncover geometric properties of algebraic varieties, can be traced back to Green's foundational papers from the 1980's. This direction of research has culminated in the formulation of Green's Conjecture, asserting that for an arbitrary algebraic curve, its intrinsic geometry (given in terms of its linear series) can be explicitly recovered from its extrinsic geometry (given in terms of equations of its canonical embedding). This deceptively simple statement has attracted a great amount of attention, and a breakthrough was achieved in 2002, when Voisin proved the conjecture for general curves of arbitrary genus. However, the case of arbitrary curves remains as challenging as ever.

Syzygies have been also used in recent years to study the birational geometry of moduli spaces of curves (with or without level structure). Often, extremal divisors on these moduli spaces can be described as jumping loci for Koszul cohomology. Their natural determinantal structure, makes these Koszul loci particularly amenable to computations, which can then be used to determine important invariants of parameter spaces, like their Kodaira dimension or cones of effective divisors.

It is a very tantalizing problem to try and extend some of these ideas to higher dimensional varieties. In one direction one would like to understand the constraints on the Betti tables of a variety coming from its geometry. In the direction of moduli spaces, it would be important to study degenerations of syzygies to singular varieties, and develop tools to computre classes of arbitrary Koszul loci in moduli.

Another major recent development in the theory of syzygies involves the notion of asymptotic regularity of an ideal. Work of Lazarsfeld, Ein, and Cutkosky connects this to interesting invariants of line bundles on varieties, and has uncovered some very surprising behavior. In work of Beheshti, Eisenbud, and Harris, it turns out that these questions are related to very classical questions from Algebraic Geometry, concerning the nature of the fibers of a general projection of a projective algebraic variety of dimension n-1 into $P^n$. These things are closely connected with the Regularity Conjecture of Eisenbud and Goto through work of Kwak and his students. Chardin and others have also made substantial studies of regularity in this period. This area is quite active.

A second development with asymptotic flavor is the very recent study (not even available in preprint form yet) by Ein and Lazarsfeld of the Betti tables of free resolutions of varieties under high Veronese embeddings (or other ``extremely ample'' embeddings). Conjectures about the case of curves were formulated by Green and Lazarsfeld long ago, and parts of them have been proven, but the higher-dimensional case has remained mysterious. It will be very useful to get the experts together in one place to explore what is possible in this direction, and that will be another goal of our workshop.

Recently there has been a surge in interest in the above areas. The workshop will be timely and will provide an opportunity to bring together researchers related to Algebraic Geometry, Commutative Algebra, and String Theory, in order to discuss exciting recent developments and ideas, and explore in new directions.

A major recent development in free resolutions came from a group of conjectures of Mats Boij and Jonas Soederberg. The original motivation of Boij and Soederberg was an attempt (now completely accomplished by these methods) to prove the Multiplicity Conjecture of Herzog, Huneke, and Srinivasan. That conjecture required some understanding of the shape of free resoluitions of graded modules over polynomial rings. The key insight of Boij and Soederberg was that it might be easier to look at the convex cone generated by the Betti tables of such resolutions than at the resolutions themselves. The original conjectures were proven by Eisenbud and Schreyer. Many other people have contributed to a very rapid development over the last several years. The current result is that we know the cone of Betti tables precisely, and also the cone of cohomology tables of vector bundles on projective spaces. We also have an understanding, only slightly less precise, of the cone of cohomology tables of coherent sheaves. The new understanding is quite orthogonal to the ways in which people had looked at these before. The area is still developing fast. Work has begun in the direction of infinite resolutions, and on other graded rings; for example, Eisenbud and Schreyer have worked on the case of cohomology tables of vector bundles on the next simplest toric variety, $P^1 x P^1$. The Boij-Soederberg theory will be another focus of our workshop.

The systematic use of syzygy techniques in order to uncover geometric properties of algebraic varieties, can be traced back to Green's foundational papers from the 1980's. This direction of research has culminated in the formulation of Green's Conjecture, asserting that for an arbitrary algebraic curve, its intrinsic geometry (given in terms of its linear series) can be explicitly recovered from its extrinsic geometry (given in terms of equations of its canonical embedding). This deceptively simple statement has attracted a great amount of attention, and a breakthrough was achieved in 2002, when Voisin proved the conjecture for general curves of arbitrary genus. However, the case of arbitrary curves remains as challenging as ever.

Syzygies have been also used in recent years to study the birational geometry of moduli spaces of curves (with or without level structure). Often, extremal divisors on these moduli spaces can be described as jumping loci for Koszul cohomology. Their natural determinantal structure, makes these Koszul loci particularly amenable to computations, which can then be used to determine important invariants of parameter spaces, like their Kodaira dimension or cones of effective divisors.

It is a very tantalizing problem to try and extend some of these ideas to higher dimensional varieties. In one direction one would like to understand the constraints on the Betti tables of a variety coming from its geometry. In the direction of moduli spaces, it would be important to study degenerations of syzygies to singular varieties, and develop tools to computre classes of arbitrary Koszul loci in moduli.

Another major recent development in the theory of syzygies involves the notion of asymptotic regularity of an ideal. Work of Lazarsfeld, Ein, and Cutkosky connects this to interesting invariants of line bundles on varieties, and has uncovered some very surprising behavior. In work of Beheshti, Eisenbud, and Harris, it turns out that these questions are related to very classical questions from Algebraic Geometry, concerning the nature of the fibers of a general projection of a projective algebraic variety of dimension n-1 into $P^n$. These things are closely connected with the Regularity Conjecture of Eisenbud and Goto through work of Kwak and his students. Chardin and others have also made substantial studies of regularity in this period. This area is quite active.

A second development with asymptotic flavor is the very recent study (not even available in preprint form yet) by Ein and Lazarsfeld of the Betti tables of free resolutions of varieties under high Veronese embeddings (or other ``extremely ample'' embeddings). Conjectures about the case of curves were formulated by Green and Lazarsfeld long ago, and parts of them have been proven, but the higher-dimensional case has remained mysterious. It will be very useful to get the experts together in one place to explore what is possible in this direction, and that will be another goal of our workshop.

Recently there has been a surge in interest in the above areas. The workshop will be timely and will provide an opportunity to bring together researchers related to Algebraic Geometry, Commutative Algebra, and String Theory, in order to discuss exciting recent developments and ideas, and explore in new directions.