Department of Mathematical Sciences

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University of Arkansas

Fayetteville, AR 72701

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E-mail: math@uark.edu

# Spring Lecture Series

**42**^{nd} Annual Spring Lecture Series

^{nd}Annual Spring Lecture Series

**Geometry and the Equations Defining Projective Varieties****March 9-11, 2017**

A series of five lectures by Robert Lazarsfeld (Stony Brook University)

**Registration is now closed.**

### Invited Participants

Daniel Erman (University of Wisconsin)

Greg Smith (Queen's University)

Wenbo Niu (University of Arkansas)

Claudiu Raicu (Notre Dame)

Lawrence Ein (University of Illinois-Chicago)

Sijong Kawk (KAIST)

Jason McCullough (Rider University)

Jinhyung Park (Korea Institute for Advanced Study)

Brooke Ullery (Harvard University)

Juergen Rathmann (Munchen, Germany)

### Organizers

Mark Johnson (University of Arkansas)

Lance Miller (University of Arkansas)

Paolo Mantero (University of Arkansas)

The principal speaker for the Lecture Series is Professor Robert Lazarsfeld, Stony Brook University. Through a series of five talks, "Geometry and the Equations Defining Projective Varieties", he will discuss the recent progress on the syzygies of smooth projective varieties.

A Public Lecture will be given by Aaron Bertam (University of Utah) on "Tropical Mathematics."**The public lecture will be taking place from 7:00pm to 8:00pm in the Reynolds Auditorium,
Room 120.**

The University of Arkansas Spring Lecture series are conferences organized every spring by the Department of Mathematical Sciences of the University of Arkansas. Each conference is focused on a specific topic chosen among the current leading research areas in Mathematics; a principal lecturer delivers a short, five-lecture course and selects a number of specialists who are invited to give talks on subjects closely related to the topic of the conference. Short talks by young Ph.D.s and finishing graduate students are solicited to complement the conference. Each Lecture Series has grown into an ideal opportunity for specialists and young researchers to meet and exchange ideas about topics at the forefront of modern mathematics.

**On Wednesday**, we will have a half day workshop for junior participants on background material
for the lecture series, starting after lunch. This will include lectures and problem
sessions lead by Wenbo Niu and Lance Miller.**The pre-conference workshop will take place Wednesday 1:30pm to 5:00pm in the Reynolds
Center, Room 117.**

The Spring Lectures is sponsored by the NSF jointly with the University of Arkansas.

**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 deg*Lα = 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
dim*X ≥ 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.

**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