Speaker: Aaron Bertram (University of Utah)
Title: [Public Lecture] Tropical Mathematics

Abstract: Mathematics rests on the pair of operations - addition and multiplication that gives us the arithmetic of numbers. Tropical mathematics replaces this with another pair of operations - maximum and addition - that also obey the basic laws of arithmetic. We can build polynomials in one and more variables based on this new arithmetic and explore the geometry (lines, conics, etc) that results. Although this geometry looks very odd, there are some uncanny similarities with "ordinary"; geometry that have inspired quite a lot of mathematics research in the past few decades, including connections with theoretical physics and string theory. In this talk we will develop some "high school"; tropical mathematics and try to give some explanation for why mathematicians are finding it so intriguing.

Speaker: Lawrence Ein (University of Illinois-Chicago)
Title: Measures of irrationalities

Abstract: We'll discuss joint work with Bastianelli, De Poi, Lazarsfeld and Ullery. We describe various measures how irrational is a projective variety. In particular, we study the case, when the variety is a very general hypersurface of high degree.

Speaker: Daniel Erman (University of Wisconsin)
Title: Syzygies of Veroneses

Abstract: I'll talk about a large-scale computation we performed to gather data about the syzygies of Veroneses. I'll explain what went into the computation and some of the conjectures that resulted from these computations. I'll also explain how these connect to the conjectures of Ein-Lazarsfeld and Ein-Erman-Lazarsfeld.
This is joint work with DJ Bruce, Steve Goldstein, and Jay Yang.

Speaker: Sijong Kwak (KAIST)
Title: The upper bound of Betti numbers in the quadratic and cubic strands.

Abstract: For irreducible, reduced projective varieties, the structure of Betti tables gives an interesting geometric meanings. In this talk, I'd like to give a talk on the upper bound and lower bound on Betti numbers with explanation of boundary cases.

Principle Speaker: Robert Lazarsfeld (Stony Brook University)
Title: Defining equations of projective varieties

Abstract: Over the past thirty years, a considerable body of work has developed loosely centered around the equations defining projective varieties. On the one hand, classical results about defining equations of curves and abelian varieties have emerged as the first cases of more general statements for higher syzygies, and the picture for general smooth varieties has started to come into focus. In terms of a natural measure of algebraic complexity, a fascinating but still mysterious dichotomy has emerged between the behavior of nonsingular varieties and arbitrary schemes. While the questions are algebraic in background, the techniques used to study them are very geometric, including for instance vanishing theorems for multiplier ideals, the geometry of Hilbert schemes and vector bundles, and Fourier-Mukai transforms on abelian varieties. The general theme of these lectures and this school will be to explore from a geometric perspective recent developments involving this circle of ideas. In more detail, consider a variety or scheme X ⊆ Pr embedded in projective space, defined by a homogeneous ideal I = IX ⊆ C[T0,...,Tr]. It is interesting to ask whether one can bound the algebraic complexity of I in terms of geometric invariants of I. The natural measure of complexity here is the CastelnuovoMumford regularity of I, which is computed by the largest degree of a generator of the syzygy modules of I: in suitable generic coordinates, this coincides with the largest degree of a generator of the initial degree of I. So the question becomes how to bound the CastelnuovoMumford regularity of X in terms of accessible geometric or algebraic data.
When X is smooth, its regularity is known to satisfy linear bounds. For example, in [3], vanishing theorems were used to show that if a smooth variety X ⊆ Pr of codimension e is cut out scheme-theoretically by hypersurfaces of degree d, then reg(X) ≤ (ed−e + 1), and this is optimal if (and only if) X is a complete intersection. This leads to a linear bound (due to Mumford [1]) in terms of deg(X), although it remains a very interesting open problem to find the best possible bound ([7], [11]). By contrast, for arbitrary schemes X defined by equations of degree d, examples of Mayr-Meyer-Bayer-Stillman [2] show that the regularity can grow as a polynomial whose degree is doubly exponential in the number of variables. A very interesting open question is to understand more clearly where the dividing line lies between these two sorts of behavior.
A second avenue of research involves the the algebraic properties of a smooth variety X embedded by the complete linear series associated to a suitably positive very ample line bundle L. A classical theorem of Castelnuovo and others asserts that if L is a line bundle of degree d ≥ 2g + 2 on a smooth curve C, then in the embedding C ⊆ Pd−g defined by L, the homogeneous ideal IC of C is generated by quadrics. In his influential papers [8], [9], Green realized that should be seen as the first cases of a much more general
3 picture for higher syzygies. For example, when deg(L) ≥ 2g + 2, then the syzygies among quadric generators Qα ∈ IC are spanned by relations of the form XLαQα = 0, where degLα = 1. This was generalized to other settings by many authors including [12], [4], the rough upshot being that as the positivity of the embedding line bundle grows, the first few modules of syzygies of the homogeneous ideal IX are generated entirely in the minimal possible degree. At the same time, it was established in [5] that when dimX ≥ 2, then "most" of the syzygies of IX have generators in as many different degrees as possible. There are several interesting conjectures in this area that would complete the general picture.
Finally, returning to the case of curves, one can consider the case of the canonical embedding C ⊆ Pg−1. A beautiful and influential conjecture of Green predicts that the resolution of the homogeneous ideal IC/Pg−1 reflects in a very precise way the presence of special divisors on C. Green's conjecture generated a huge amount of activity, leading up to the spectacular work [13], [14] of Voisin establishing it for general curves via a study of Hilbert schemes of K3 surfaces. More recently, it was observed in [6] that a variant of Voisin's ideas lead to a surprisingly simple proof of the gonality conjecture of [10], asserting that one can read off the gonality of a curve from the resolution of its ideal under the embedding defined by any one line bundle of sufficiently large degree. It remains a very interesting project to see whether these Hilbert scheme ideas can be further exploited.

Speaker: Jason McCullough (Rider University)
Title: Rees-Like Algebras and the Eisenbud-Goto Conjecture

Abstract: Regularity is a measure of the computational complexity of a homogeneous ideal in a polynomial ring. There are examples in which the regularity growth is doubly exponential in terms of the degrees of the generators but better bounds were conjectured for 'nice'; ideals. Together with Irena Peeva, I discovered a construction that overturned some of the conjectured bounds for 'nice' ideals - including the long-standing Eisenbud-Goto conjecture. Our construction involves two new ideas that we believe will be of independent interest: Rees-like algebras and step-by-step homogenization. I'll explain the construction and some of its consequences.

Speaker: Wenbo Niu (University of Arkansas)
Title: Projective normality and syzygies for some nonsingular varieties.

Abstract: In this talk, we will discuss problems related to projective normality and higher syzygies for powers of line bundles on nonsingular projective varieties. We focus on two situations: powers of ample line bundles on Calabi-Yau varieties and pluricanonical divisors on varieties of general type. These two cases follow the same approach which is to consider how the Arbarello-Sernesi module associated to the variety is generated as a graded module, which can be further reduced to considering surjecitivity of multiplication maps of line bundles.

Speaker: Jinhyung Park (Korea Institute for Advanced Study)
Title: Castelnuovo-Mumford regularity bounds for projective varieties

Abstract: It is well known that the Castelnuovo-Mumford regularity of an embedded projective variety gives an upper bound for the degree of minimal defining equations. There is a very famous conjecture, due to Eisenbud and Goto, that the Castelnuovo-Mumford regularity is bounded by degree - codimension + 1. This conjecture is equivalent to giving sharp bounds for both Castelnuovo-Mumford regularity of structure sheaf and normality. First, I show a sharp bound for Castelnuovo-Mumford regularity of structure sheaf of a smooth variety, and classify the extremal and the next to extremal cases. On the other hand, the conjectured bound for normality (of smooth varieties) remains widely open. In the remaining time, I discuss about various approaches to show certain normality bounds.
This talk is based on joint work with Sijong Kwak.

Speaker: Claudiu Raicu (Notre Dame University)
Tile: Regularity and cohomology of determinantal thickenings

Abstract: I will explain how to determine the cohomology groups of arbitrary equivariant thickenings of generic determinantal ideals, as well as the ranks of the maps in cohomology induced by inclusions of such thickenings. As an application, I will give a concrete description of the linear functions that compute the Castelnuovo-Mumford regularity of sufficiently large powers and symbolic powers of determinantal ideals.

Speaker: Juergen Rathmann (Munchen, Germany)
Title: An effective bound for the gonality conjecture

Abstract: The gonality of an algebraic curve is the minimal degree of a base point free linear system. Ein and Lazarsfeld showed that the gonality can be recovered from the shape of the resolution of any embedding of sufficiently large degree ("gonality-conjecture" of Green and Lazarsfeld). I want to explain that the conjecture already holds for embeddings of degree at least 4g−3.

Speaker: Greg Smith (Queen's University)
Title: Locally-Free Resolutions of Toric Vector Bundles

Abstract: To each torus-equivariant vector bundle over a smooth complete toric variety, we associated a representable matroid (essentially a finite collection of vectors). In this talk, we will describe how the combinatorics of the matroid encodes a locally-free resolution of the vector bundle. With some luck, we will also describe some applications to the equations and syzygies of projective toric varieties.

Speaker: Brooke Ullery (Harvard University)
Title: Normality of secant varieties

Abstract: If X is a smooth variety embedded in projective space, we can form a new variety by looking at the closure of the union of all the lines through two points on X. This is called the secant variety of X. Similarly, the Hilbert scheme of two points on X parametrizes all length 2 zero-dimensional subschemes. I will talk about how these two constructions are related. More specifically, I will show how we can use certain tautological vector bundles on the Hilbert scheme to help us understand the geometry of the secant variety, leading to a proof that for sufficiently positive embeddings of X, the secant variety is a normal variety.

Speaker: Yuan Wang (University of Utah)
Title: Characterization of Abelian Varieties for log pairs

Abstract: Let X be a projective variety and ∆ an effective Q-divisor on X. A celebrated theorem of Kawamata says that if X is smooth and κ(X) = 0, then the Albanese morphism of X is an algebraic fiber space. Later it was shown by Zhang that if (X,∆) is a log canonical pair and −(KX + ∆) is nef, then the Albanese morphism of any smooth model of X is an algebraic fiber space. In this talk, I will further discuss the relationship between κ(KX+∆) = 0, positivity of −(KX + ∆), and the Albanese map of X, and present some related results and examples. In particular, I will present a result that generalizes Kawamata's result to log canonical pairs.

Speaker: Ruijie Yang (Stony Brook University)
Title: Higher syzygies on surfaces via the geometry of nested Hilbert schemes of points

Abstract: In this talk, we will discuss higher syzygies of adjoint linear series on smooth projective surfaces. We first calculate the nef cone of the nested Hilbert schemes of points on several classes of surfaces. Then we will show how these calculations with Voisin's method can give effective bounds along the lines of Mukai's conjecture for those surfaces.
This is joint work in progress with Tim Ryan.

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 References

[1] D. Bayer and D. Mumford, What can be computed in algebraic geometry?, in Computational Algebraic Geometry and Commutative Algebra (Eisenbud and Robbiano, eds.) Cambridge Univ. Press, 1993, pp. 1 – 48.
[2] D. Bayer and M. Stillman, On the complexity of computing syzygies, J. Symbolic Computation 6 (1988), 135–147.
[3] Aaron Bertram, Lawrence Ein and Robert Lazarsfeld, Vanishing theorems, a theorem of Severi, and the equations defining projective varieties, J. AMS 4 (1991), pp. 587–602.
[4] Lawrence Ein and Robert Lazarsfeld, Syzygies and Koszul cohomology of smooth projective varieties of arbitrary dimension, Invent. Math. 111 (1993), 51–67.
[5] Lawrence Ein and Robert Lazarsfeld, Asymptotic syzygies of algebraic varieties, Invent. Math. 190 (2012), 603–646.
[6] Lawrence Ein and Robert Lazarseld, The gonality conjecture on the syzygies of algebraic curves of large degree, to appear in Publ. Math. IHES.
[7] David Eisenbud and S. Goto, Linear free resolutions and minimal multiplicity, J. Alg. 88 (1984), 89–133.
[8] Mark Green, Koszul cohomology and the geometry of projective varieties, J. Diff. Geom. 19 (1984), 125–171.
[9] Mark Green, Koszul cohomology and the geometry of projective varieties, II, J. Diff. Geom. 20 (1984), 279–289.
[10] Mark Green and Robert Lazarsfeld, On the projective normality of complete linear series on an algebraic curve, Invent. Math. 83 (1985), 73 – 90.
[11] Sijong Kwak, Castelnuovo-Mumford regularity for smooth subvarieties of dimensions 3 and 4, J. Alg. Geom 7 (1988), 195–206.
[12] Giuseppe Pareschi, Syzygies of abelian varieties, Journal of the AMS 13 (2000), 651– 664.
[13] Claire Voisin, Green's generic syzygy conjecture for curves of even genus lying on a K3 surface, J. Eur. Math. Soc. 4 (2002), 363–404.
[14] Claire Voisin, Green's canonical syzyzy conjecture for generic curves of odd genus, Compos. Math. 141 (2005), 1163–1190.

For this lecture series, only talks by the main speaker were recorded.

SLS 2017 Video Archive Playlist

Robert Lazarsfeld
Lecture 1
SLS 2017 - 01

Robert Lazarsfeld
Lecture 2
SLS 2017 - 02

Robert Lazarsfeld
Lecture 3
SLS 2017 - 03

Robert Lazarsfeld
Lecture 4
SLS 2017 - 04

Robert Lazarsfeld
Lecture 5
SLS 2017 - 05

SLS Problems List [PDF]