- Daniele Agostini, Tuebingen
Ulrich sheaves for secants of curves of high degree
We show how to construct Ulrich sheaves of rank one on the higher secants of high degree curves, via
bundles on the symmetric product of a curve. This is joint work with Mario Kummer and Jinhyung Park.
- Marian Aprodu, Bucharest
Resonance and vector bundles
Resonance varieties are algebro-geometric objects that emerged from the geometric group theory. They are naturally associated to vector subspaces in second exterior powers and carry natural scheme structures that can be non-reduced. In algebraic geometry, they made an unexpected appearance in connection with syzygies of canonical curves. In this talk, based on works with G. Farkas, C. Raicu, A. Suciu, and J. Weyman I report on some recent results concerning the geometry of resonance schemes in the vector bundle case.
- Aldo Conca, Genova
Two Bounds on Castelnuovo-Mumford Regularity
I will present two results concerning bounds on the Castelnuovo-Mumford regularity. The first focuses on ideals with polynomial parametrization, a joint work with Francesca Cioffi. The second addresses ideals associated with general subspace arrangements, developed in collaboration with Manolis Tsakiris.
- Alastair Craw, Bath
Hilbert schemes of points on canonical surfaces
For a surface S with canonical (resp. symplectic) singularities and for any positive integer n, I'll explain why the Hilbert scheme of n-points on S also has canonical (resp. symplectic) singularities. In particular, these Hilbert schemes are normal varieties. The main results generalise well-known work of Fogarty and Beauville for non-singular surfaces. Our main tool is a generalisation of the Le Bruyn-Procesi theorem that describes the invariant algebra for the natural action of the product of general linear groups on the space of representations of a quiver for a given dimension vector. This is joint work with Ryo Yamagishi.
- Daniel Erman, Honolulu
Multigraded syzygies
I will discuss some new results about derived categories of toric varieties, and the implications they have for the study of multigraded syzygies.
- Sara Filippini, Lecce
Residual intersections and Schubert varieties
The notion of residual intersections was introduced by Artin and Nagata. Let I be an ideal in a local Cohen-Macaulay ring R, and A = (a_1, \ldots, a_s) \subsetneq I. Then J = A:I is called an s-residual intersection of I if ht(J) \geq s \geq ht(I). Residual intersections provide a generalization of linkage. Indeed, if J = A:I and I = A:J for A a regular sequence, I and J are said to be linked.
I will show how results of Huneke and of Kustin and Ulrich on residual intersections for standard determinantal ideals and Pfaffian ideals arise in the context of ideals of Schubert varieties in the big opposite cell of homogeneous spaces. This is joint work with X. Ni, J. Torres and J. Weyman.
- Lorenzo Guerrieri, Kraków
Licci ideals of codimension 3 and Schur functors
Let R be a regular local (or graded) ring. An ideal
of R is said to be licci if it is in the linkage
class of a complete intersection. Licci ideals of
codimension 1 or 2 coincide with perfect ideals. In
codimension 3 the problem of classifying licci
ideals among perfect ideals is wide open.
In this talk we show how the licci ideals of
codimension 3 can be related to certain Schur
functors, and their linkage properties can be
deduced by combinatorial methods. The main
application provides a constructive algorithm to
describe the licci ideals of codimension 3 defining
rigid algebras. These ones represent generic models
for all licci ideals of codimension 3.
- Xianglong Ni, Berkeley
Herzog classes of grade three licci ideals
By work of Buchweitz and Herzog, there is a well-defined classification of licci ideals up to deformation. The equivalence classes obtained in this manner are called Herzog classes. For grade 2 perfect ideals (all of which are licci) the Herzog class of I is determined by its minimal number of generators, i.e. the vector space dimension of I/mI. Furthermore, the class of a linked ideal K:I can be inferred from the dimension of the subspace (K+mI)/mI. Assuming equicharacteristic zero, we generalize this to grade 3 licci ideals, where we can describe all Herzog classes with the assistance of representation theory. In this setting, the class of K:I depends on the incidence of (K+mI)/mI with a distinguished partial flag on I/mI. This is based on ongoing joint work with Lorenzo Guerrieri and Jerzy Weyman.
- Jinhyung Park, Daejeon
Effective gonality theorem on weight-one syzygies of algebraic curves
In 1986, Green-Lazarsfeld raised the gonality conjecture asserting that the gonality gon(C) a smooth projective curve C of genus g can be read off from weight-one syzygies of a sufficiently positive line bundle L, and also proposed possible least degree of L, that is 2g+gon(C)-1. In 2015, Ein-Lazarsfeld proved the conjecture when deg(L) is sufficiently large, but the effective part of the conjecture remained widely open and was reformulated explicitly by Farkas-Kemeny a few years ago. We show an effective vanishing theorem for weight-one syzygies, which implies that the gonality conjecture holds if deg(L) is at least 2g+gon(C) or equal to 2g+gon(C)-1 and C is not a plane curve. As Castryck observed that the gonality conjecture may not hold for a plane curve when deg(L)=2g+gon(C)-1, this result is the best possible and thus gives a complete answer to the gonality conjecture. This is joint work with Wenbo Niu.
- Claudia Polini, Notre Dame
Syzygies of the residue field of a Golod ring
We prove a surprising finiteness result for Golod rings. All the syzygies of the residue field of a local Golod ring are sums of a finite set of indecomposable modules. This finitistic behaviour is diametrically opposed to the case of Gorenstein rings, where each syzygy of the residue field is indecomposable. In addition we describe in detail the case of a local ring of embedding dimension two that is not a complete intersection. In this case, all syzygy modules are direct sums of only three possible modules, the residue field, the maximal ideal, and the dual of the maximal ideal.
- Claudiu Raicu, Notre Dame
Cohomology of line bundles on the incidence correspondence
A fundamental problem at the confluence of algebraic geometry, commutative algebra and representation theory is to understand the structure and vanishing behavior of the cohomology of line bundles on (partial) flag varieties. Over fields of characteristic zero, this is the content of the Borel-Weil-Bott theorem and is well-understood, but in positive characteristic it remains wide open, despite important progress over the years. In my talk I will describe recent developments in the case of the incidence correspondence - the partial flag variety consisting of pairs of a point in projective space and a hyperplane containing it.
- Ritvik Ramkumar, Cornell
Hilbert scheme of points on threefolds
The Hilbert scheme of points on a smooth threefold is an important moduli space with many open questions regarding its singularities. In this talk, I will focus on describing the structure of the smooth points of this Hilbert scheme. Time permitting, I will also discuss the mildly singular points of this Hilbert scheme; in particular, their tangent space and singularities. This is all joint (ongoing) work with Joachim Jelisiejew and Alessio Sammartano.
- Kristian Ranestad, Oslo
Varieties of apolar schemes to powers of quadrics
I shall report on recent work with Grzegorz and Michal Kapustka on the variety of apolar schemes of minimal length to a power of a quadric. We determine the minimal length (the cactus rank) of any such form.
In the case of a third power of a ternary quadric we show that the variety is a tangent developable of a rational normal curve, and in the case of a square of a quaternary quadric the variety has three five dimensional components. In both cases the cactus rank of the quadric power is ten, while the rank is eleven. The main focus of our approach is the syzygies of the finite length apolar schemes.
- Alessio Sammartano, Milano
Irrational components of Hilbert schemes of points
The Hilbert scheme of points in affine n-dimensional space parametrizes finite subschemes of a given length. It is smooth and irreducible if n is at most 2, singular and reducible if n is at least 3. Understanding its irreducible components, their singularities and birational geometry, has long been an inaccessible problem. In this talk, I will describe substantial progress on this problem achieved in recent years. In particular, I will focus on the problem of rationality of components. This is based on a joint work with Gavril Farkas and Rahul Pandharipande.
- Hal Schenck, Auburn
Syzygies of permanental ideals
We describe the minimal free resolution of the ideal of 2×2 subpermanents of a 2×n generic matrix M. In contrast to the case of 2×2 determinants, the 2×2 permanents define an ideal which is neither prime nor Cohen-Macaulay. We combine work of Laubenbacher-Swanson on the Gröbner basis of an ideal of 2×2 permanents of a generic matrix with our previous work connecting the initial ideal of 2×2 permanents to a simplicial complex. The main technical tool is a spectral sequence arising from the Bernstein-Gelfand-Gelfand correspondence. Joint work with F. Gesmundo, H. Huang, J. Weyman.
- Gregory G. Smith, Queen's
Smooth Quot schemes
How can we understand all coherent sheaves on a projective scheme? Quot schemes offer a geometric answer by parameterizing the sheaves, and the Hilbert polynomial gives a natural stratificaiton into disjoint pieces. We will survey the key features of these parameter spaces for the quotients of a finite direct sum of the structure sheaf on projective space. Moreover, we will complete classify the smooth Quot schemes of this form. This talk is based on joint work with Roy Skjelnes and Mike Stillman.
- Emanuele Ventura, Torino
Border apolarity and varieties of sums of powers
Border tensor rank has a remarkable role in theoretical computer science,
especially in the estimation of the exponent of matrix multiplication.
Buczyńska and Buczyński introduced border varieties of sums of powers
to encode border decompositions of a tensor. These are objects in the Haiman-Sturmfels
multigraded Hilbert scheme and are the border counterparts of the classical
varieties of sums of powers in the Hilbert scheme.
In this talk, I shall describe examples and results in the context of tensors and homogeneous
polynomials, mentioning the role of multigraded regularity. This is based on joint work with Tomasz Mańdziuk.
