Syzygies in Algebra and Geometry 2015
Castelnuovo-Mumford regularity and Ratliff-Rush closure
Speaker: N.V. Trung
Abstract
We establish relationships between the Castelnuovo-Mumford regularity of
standard graded algebras and the Ratliff-Rush closure of ideals. These
relationships can be used to compute the Ratliff-closure and the regularities
of the Rees algebra and the fiber ring. As a consequence, these regularities
are equal for large classes of monomial ideals in two variables, thereby
confirming a conjecture of Eisenbud and Ulrich for these cases. This is a joint work with M.E. Rossi and D.T. Trung.
On syzygies of ruled surfaces
Speaker: Euisung PARK
Abstract
In this talk, I will speak about the minimal free resolution of homogeneous
coordinate rings of a ruled surface $S$ over a curve of genus $g$
with the numerical invariant $e<0$ and a minimal section $C_0$. Let
$L\in \mbox{Pic} X$ be a line bundle in the numerical class of
$aC_0+bf$ such that $a \geq 1$ and $2b-ae=4g-1+k$ for some $k\ge
{\rm max}(2,-e)$. We prove that the Green-Lazarsfeld index ${\rm
index}(S,L)$ of $(S,L)$, i.e. the maximum $p$ such that $L$
satisfies condition $N_{2,p}$, satisfies the inequalities
\begin{equation*}
\frac{k}{2}-g \leq {\rm index}(S,L) \leq \frac{k}{2} -\frac{ae+3}{2}
+{\rm max} \big( 0,\left\lceil \frac{2g-3+ae-k}{4}\right\rceil
\big).
\end{equation*}
Also if $S$ has an effective divisor $D\equiv 2C_0+e\mathfrak f$,
then we obtain another upper bound of ${\rm index}(S,L)$, i.e.,
${\rm index}(S,L)\leq k +{\rm max} \big( 0,\left\lceil
\frac{2g-4-k}{2}\right\rceil \big)$. This gives a better bound in
case $b$ is small compared to $a$. Finally, I will show that for each $e \in \{
-g,\ldots , -1 \}$ there exists a ruled surface $S$ with the
numerical invariant $e$ and a minimal section $C_0$ which has an
effective divisor $D\equiv 2C_0+e\mathfrak f$.
Cones of Betti tables and Hilbert functions
Speaker: Mats Boij
Abstract
Studying the possible Betti tables of graded modules, it turned
out to be useful to relax the question and only look at the Betti tables up to
scaling, i.e., to study the cone spanned by the Betti tables in a suitable
vector space. In the standard graded case, Macaulay's theorem gives us a
complete classification of Hilbert functions of cyclic modules, but in other
cases we are lacking such a classification and results on cones of Hilbert
functions are useful. We can also combine the two questions and look at
Hilbert functions of modules with some properties seen from the Betti table
but not directly on the Hilbert function, as in the case of modules with
bounded regularity. I will give a survey on some of the work that has been
done on cones of Betti tables and Hilbert functions over the last few years
including some recent joint work with Gregory G. Smith.
Dual graphs of projective schemes
Speaker: Matteo Varbaro
Abstract
Given a projective scheme $X$, its dual graph $G(X)$ is the graph
whose vertices are given by the irreducible components of $X$, and such that 2
vertices are connected by an edge iff the intersection of the 2 correspondent
components is a codimension 1 subscheme of $X$. A classical result of
Hartshorne says that, if $X$ admits an arithmetically Cohen-Macaulay (aCM)
embedding, then $G(X)$ is connected. In a joint work with Bruno Benedetti and
Barbara Bolognese, we improved the conclusion assuming that $X$ admits an
arithmetically Gorenstein embedding: in this case, if the (Castelnuovo-Mumford) regularity of $X$ (in such an embedding) is $r+1$, and the regularity of each irreducible component of $X$ is $\leq d$, then $G(X)$ is $[(r+d-1)/d]$-connected. We also proved that for any graph there exists a
reduced projective curve $C$ admitting an aCM embedding in which all the
irreducible components of C are rational normal scrolls. During the talk we
will discuss these features, some examples, and open questions on dual
graphs.
Cohomology of thickenings of projective varieties
Speaker: Anurag K. Singh
Abstract
Let $X$ be a smooth projective subvariety of $\mathbb{PP}^n$
over a field of characteristic zero. We discuss a version of the Kodaira
vanishing theorem for thickenings of $X$ in $\mathbb{PP}^n$, and a
related result on the injectivity of the natural maps from Ext modules to
local cohomology modules.
This is a joint work with Bhatt, Blickle, Lyubeznik, and Zhang.
The syzygies of some thickenings of determinantal varieties
Speaker: Claudiu Raicu
Abstract
The space of m x n matrices admits a natural action of the group
GL_m x GL_n via row and column operations on the matrix entries. The
invariant closed subsets are the determinantal varieties defined by the
<<(reduced) ideals of minors of the generic m x n matrix. The minimal free
resolutions for these ideals are well-understood by work of Lascoux and
others. There are however many more invariant ideals which are non-reduced,
and whose syzygies are quite mysterious. These ideals correspond to
nilpotent structures on the determinantal varieties, and they have been
completely classified by De Concini, Eisenbud and Procesi. In my talk I will
recall the classical description of syzygies of determinantal varieties, and
explain how this can be extended to a large collection of their thickenings.
Joint work with Jerzy Weyman.
Cayley-Chow forms of $K3$ surfaces and Ulrich bundles
Speaker: Marian Aprodu
Abstract
An Ulrich bundle on a projective variety is a vector bundle that
admits a completely linear resolution as a sheaf on the projective space.
Ulrich bundles are semi-stable and the restrictions to any hyperplane
section remain semi-stable. This notion originates in classical algebraic
geometry, being related to the problem of finding, whenever possible,
linear determinantal or linear pfaffian descriptions of hypersurfaces in a
complex projective space. Generally, the existence of an Ulrich bundle has
nice consequences on the equations of the given variety, specifically, the Cayley-Chow form is the determinant of a matrix of linear forms in the
Pluecker coordinates. We prove existence of stable rank-two Ulrich bundles
on polarized $K3$ surfaces with a mild Brill-Noether condition. As a
consequence, we obtain an explicit pfaffian representation of the
associated Cayley-Chow form.
This is a joint work with Gavril Farkas and Angela Ortega.
TBA
Speaker: Daniele Faenzi
Abstract
TBA
Betti Tables of ND(1)-schemes
Speaker: Jeaman Ahn
Abstract
In this talk, we introduce ND(1)-schemes, which generalize the
concept of `being nondegenerate' from the case of varieties to the case of
more general closed subschemes and give a geometric interpretation on Betti
numbers of ND(1)-schemes. For this purpose, we use elimination mapping
cone theorem and generic initial ideal theory. We also provide some
illuminating examples of our results via calculations done with Macaulay 2.
Syzygies of curves via $K3$ surfaces
Speaker: Michael Kemeny
Abstract
In this talk I will explain how polarised $K3$ surfaces can be
used to study the syzygies of (general) embedded curves. In
particular, such methods have recently been used to resolve two
conjectures which can be considered as analogues of Green's conjecture
on the minimal free resolution of the ideal sheaf of a canonically
embedded curve. The first result is a proof of the Prym-Green
conjecture for Prym-canonically embedded curves of odd genus; i.e. the
Prym-canonical embeddeding of a general such curve has a natural
resolution. The second result is a proof of the generic case of a
famous conjecture of M. Green and R. Lazarsfeld relating property Np
to secants of embedded curves. If time permits I will also discuss
recent progress on the higher level analogue of the Prym-Green
conjecture. All results are joint with G. Farkas.
On some recent consequences of Serre's condition $S_l$
Speaker: Hailong Dao
Abstract
Serre's condition $S_2$ is quite familiar to commutative algebraists as part of the condition for normality. On the surface it appears to be a rather weak condition. In this talk we will discuss some surprising consequences of $S_l$ for $l=2$ and higher. These consequences involve cohomological dimension, depth, $h$-vector, and in the square-free monomial case, the Castelnuovo-Mumford regularity of relevant ideals. We will also discuss some intriguing open questions on the tightness of these statements. This talk is based on recent joint work and discussions with D. Eisenbud, K. Han, S. Takagi, and M. Varbaro.
The minimal graded free resolution of a star-configuration in $\mathbb{P}^n$ and secant varieties
Speaker: Yong-Su Shin
Abstract
For positive integers $r$ and $s$ with $1\leq r \leq \min \{n,s\}$, suppose $F_1,\ldots,F_s$ are general forms in $R = \mathbb{k}[x_0,x_1, \ldots , x_n]$ of degrees $d_1,\ldots,d_s$, respectively.
We call the variety $X$ defined by the ideal
$\cap_{1\leq i_1\lt\ldots\lt i_r\leq s} (F_{i_1},\ldots,F_{i_r} )$ a star-conguration in $\mathbb{P}^n$ of type
$(r,s)$. In particular, if $F_1,\ldots,F_s$ are
general linear forms in $R$, then we call $X$ a linear star-conguration in $\mathbb{P}^n$ of type $(r,s)$. As an application, we introduce secant varieties to the varieties of reducible forms.
Koszul algebras and their homological properties
Speaker: Aldo Conca
Abstract
Koszul algebras are certain algebras defined by quadratic relations.
Not all quadratic algebras are Koszul but most of the quadratic algebras that arise naturally are in fact Koszul.
Unfortunately there is no finite criterion, no algorithm, to test Koszulness. In my talk I will explain some special features of syzygies of and over Koszul algebras.
The talk is based on two recent papers:
Avramov, Luchezar L.; Conca, Aldo; Iyengar, Srikanth B.
Subadditivity of syzygies of Koszul algebras
Math. Ann. 361 (2015), no. 1-2, 511--534.
arXiv:1308.6811
and
Conca, Aldo, Iyengar, Srikanth B., Nguyen Hop, Romer, Tim,
Absolutely Koszul algebras and the Backelin-Roos property,
Acta Mathematica Vietnamica 2015
arXiv:1411.7938
Contributed talks
On the Koszul property for numerical semigroup rings
Speaker : Dumitru Stamate (University of Bucharest & IMAR)
Abstract
Let $H$ be a numerical semigroup. We give effective bounds for its multiplicity $e(H)$ such that $gr(K[H])$ is Koszul.
We conjecture that not all the values in the range are possible, and this correlates to a series of conjectures of Eisenbud, Green and Harris on a Generalized Cayley-Bacharach statement.
We describe the Koszul property for several classes of numerical semigroups and we study the relationship with the Cohen-Macaulay property of the $gr(K[H])$.
Joint work in progress with Juergen Herzog.
Varieties of powers and the Fr$\ddot{o}$berg conjecture
Speaker : Alessandro Oneto (Stockholm University)
Abstract
Motivated by a Waring problem for higher degree forms, i.e. on additive decomposition of degree $kd$
homogeneous polynomials as sum of $k$-th powers, we study the secant varieties of varieties parametrizing $k$-th powers
of degree $d$ forms. This is related to the Fr$\ddot{o}$berg conjecture on the Hilbert series of generic ideals.
In this talk, we want to describe the relation between the two problems and the current situation about these questions.
It is mainly based on joint works with R. Fr$\ddot{o}$berg, G. Ottaviani and B. Shapiro.
Ayesha Asloob Qureshi (Osaka University)
Toric rings assocaited with isotone maps between posets
Speaker : Ayesha Asloob Qureshi (Osaka University)
Abstract
Our main goal is to study the toric ring $K[P,Q]$ which is generated by the monomials arising from isotone maps from $P$ to $Q$. Such rings are a natural generalization of the classical Hibi rings. We investigate the conditions on $P$ and $Q$ to determine when defining ideal of $K[P,Q]$ is quadratically generated and when it has square-free generators. Also, we compute the dimension of $K[P,Q]$.