Speaker

Affiliation

Title of 3 lectures

Ngo Viet Trung

Institute of Mathematics, Hanoi

Title: Partial Castelnuovo-Mumford regularity
Abstract:  The partial Castelnuovo-Mumford regularity is defined by the vanishing degree of some local cohomology modules or of some syzygies. In these lectures I will present an ideal-theoretical characterization of the partial Castelnuovo-Mumford regularities and use it to prove several results on the behaviour of powers of homogeneous ideals.

Tony J. Puthenpurakal

 IIT Bombay, Mumbai, India

Title: Itoh's conjecture for integral closure filtration
Abstract:Let A be an analytically unramified Cohen-Macaulay local ring of dimension greater than or equal to 3. Let I be a zero-dimensional ideal in A. We show that if the third Hilbert coefficient of the integral closure filtration F of I vanishes then the associated graded ring of  F  is Cohen-Macaulay.

Tai Huy Ha

Tulane University,

New Orleans

Title: Binomial expansion of powers of sums of ideals
Abstract: Let A and B be algebras over a field k and let I and J be ideals in A and B, respectively. We shall discuss when symbolic powers, integral closures of powers, and rational powers of the sum I+J (considered as an ideal in A \otimes_k B) has a binomial expansion in terms of corresponding powers of I and J.

Bernd Ulrich

Purdue University,

West Lafayette

Title: Linkage, residual intersection, and applications
Abstract: Linkage, or liaison, is a tool for classifying and studying varieties and ideals that has its origins in 19th century algebraic geometry. Its generalization, residual intersection, has broad applications in enumerative geometry, intersection theory, the study of Rees rings, and multiplicity theory. After surveying basic properties of linkage, we will focus on the computation of Picard groups and divisor class groups and on the structure of rigid algebras in the linkage class of a complete intersection. We will describe applications of residual intersections and explain the techniques used to determine their Cohen-Macaulayness, canonical modules, duality properties, and defining equations. An emphasis will be on weakening the hypotheses classically required in this subject.

 Linkage, or liaison, is a tool for classifying and studying varieties and ideals that has its origins in 19th century algebraic geometry. Its generalization, residual intersection, has broad applications in enumerative geometry, intersection theory, the study of Rees rings, and multiplicity theory. After surveying basic properties of linkage, we will fo- cus on the computation of Picard groups and divisor class groups and on the structure of rigid algebras in the linkage class of a complete intersection. We will describe ap- plications of residual intersections and explain the techniques used to determine their Cohen-Macaulayness, canonical modules, duality properties, and defining equations. An emphasis will be on weakening the hypotheses classically required in this subject.

Claudia Polini

University of Notre Dame, Notre Dame

Title: The module of differentials and the module of derivations and their applications
Abstract: This series of 3 talks is concerned with the module of differentials and the module of derivations and their applications. In the first talk we will introduce the module of differentials and derivations and we will discuss their basic properties. In the second talk we will discuss the sum of all links of an ideal, the Jacobian ideal, and the differents, which define ramification loci. We express these objects in terms of the last nonzero comparison map between a Koszul complex and a free resolution of an ideal.

We establish connections between these objects and provide an effective method for computing them. Our techniques include residual intersections and linkage theory. In particular we obtain interesting formulas for determinantal ideals of generic matrices and perfect Gorenstein ideals of height three. In the third talk we will describe the structure of the module of derivations and its connections with singularities and vector fields of varieties. Modules of derivations are not well understood -- despite great advances on the Zariski-Lipman conjecture, there is still no complete characterization for when they are free.

Our work is partially motivated by a question of Poincaré, who asked how to decide whether a polynomial vector field in the complex plane leaves some algebraic curve invariant. We reformulate this problem in terms of bounding from below the initial degree of the module consisting of all vector fields that leave a fixed curve invariant. This module is a quotient of the module of derivations.

Le Tuan Hoa

Institute of Mathematics, Hanoi

Title:Stability index of the Castelnuovo-Mumford regularity
Abstract: Let I be a non-zero proper homogeneous ideal of a polynomial ring. A celebrated theorem of Cutkosky-Herzog-Trung and Kodiyalam states that the Castelnuovo -Mumford regularity reg(I^n) is equal to dn + a for some positive integer d and a non-negative integer a. The smallest positive integer t from which reg(I^n) = dn + a for all n >= t is called the stability index of the Castelnuovo -Mumford regularity of I and denoted by reg-stab(I). A similar statement also holds for the integral closures of I^n, and the corresponding stability index is called normal stability index of the Castelnuovo-Mumford regularity of I. In these lectures I will provide some recent results on bounding these indexes with the mail focus on monomial ideals.

Anurag Singh

University of Utah,

Salt Lake City

Title: When are the natural embeddings of classical invariant rings pure?
Abstract: Consider a reductive linear algebraic group G acting linearly on a polynomial ring S over an infinite field; key examples are the general linear group, the symplectic group, the orthogonal group, and the special linear group, with the classical representations as in Weyl's book: for the general linear group, consider a direct sum of copies of the regular representation and copies of the dual; in the other cases take copies of the regular representation.  The invariant rings in the respective cases are determinantal rings, rings defined by Pfaffians of alternating matrices, symmetric determinantal rings, and the Plücker coordinate rings of Grassmannians; these are the classical invariant rings of the title, with S^G in S being the natural embedding.

A subring R of a ring S is pure if the inclusion remains injective upon tensoring with an arbitrary R-module.  Over a field of characteristic zero, a reductive group is linearly reductive; it follows that the invariant ring S^G is a direct summand of S as an S^G-module, equivalently that S^G is a pure subring of S.  Over fields of positive characteristic, reductive groups are typically no longer linearly reductive.  We determine, in the positive characteristic case, precisely when the inclusion of S^G in S is pure.  This is joint work with Melvin Hochster, Jack Jeffries, and Vaibhav Pandey.

Along the way, we will discuss connections of purity with split and solid extensions, techniques used in our proofs including principal radical systems, work of Kraft and Schwarz on nullcones, and also some open questions.

Marc Chardin

CNRS & Sorbonne Université, Paris

Title: Image and fibres of a rational map
Abstract: These three lectures will provide an introduction to works on the determination of the image and the fibers of a rational map using syzygies. This line of work was initially motivated by the interplay between parametrized and implicit representations of a surface (or a curve) in geometric modelling. One of the main tools, in the algebraic approach that I will present, is the use of the symmetric and Rees algebras of an ideal and the study of the kernel of the natural map between these. It is intimately linked to results about powers of graded ideals. On the computational side that geometric modelling is using, it gives rise to representations of a surface (or other rational varieties) that are in connection to both the implicit and the parametrized versions.

Steven Dale Cutkosky

University of Missouri, Columbia

Title:Multiplicities, intersection theory and volumes of filtrations
Abstract: In these lectures we discuss the generalization of some classical results on ideals to (not necessarily Noetherian) filtrations. The multiplicity of an m-primary ideal extends to the multiplicity of a filtration of m-primary ideals. This multiplicity can be computed as the volume of a slice of the cone generated by a suitable semigroup (an Okounkov body). This will be explained in the first lecture. In the case of a divisorial filtration, this volume can be interpreted geometrically as an intersection multiplicity. Many classical theorems on multiplicities of ideals extend to filtrations. For instance, Rees' theorem characterizing when ideals I contained in J have the same multiplicity extends to filtrations.

The mixed multiplicities of m-primary ideals also extend to filtrations of m-primary ideals. In the case of divisorial filtrations, these mixed multiplicities can also be interpreted geometrically as intersection multiplicities.  Classical theorems on mixed multiplicities of ideals extend to filtrations, including the Minkowski inequalities and characterization of equality.

Other topics that will be covered are the concept of the analytic spread of a filtration, which is computed from the Rees algebra of the filtration, and more general multiplicities of filtrations, such as the epsilon multiplicity.

Ken-ichi Yoshida

Nihon University, Nihon

Title: Hilbert-Kunz multiplicities of integrally closed ideals
Abstract:  Let (A,m) be a Noetherian unmixed local ring containing a field of prime characteristic p. In 1969, Kunz defined the Hilbert-Kunz multiplicity e_HK(m) of the maximal ideal m and proved e_HK(m)=1 iff A is regular.   In 2000, Watanabe and I proved the converse, that is, if A is an unmixed local ring of e_HK(m)=1 then A is regular. Thus if A is a non-regular, unmixed local ring, then e_HK(m) > 1. In 2005, Watanabe and I posed a conjecture on lower bounds on e_HK(m) for non-regular local ring A based on Monsky's observation.   Several authors have studied on this conjecture but it has not been settled yet.
 First, I will introduce several answers on this conjecture.   Next, I will discuss about the difference_HK(I)-\ell_A(A/I) for m-primary integrally closed ideals I in a 2-dimensional normal local domain. In 2012, Celikbas et.al.  proved that this difference is always non negative. Note that the difference is not necessarily non-negative in general. Moreover, Watanabe and I (2001) gave a formula for good ideals using classical McKay correspondence in a 2-dimensional quotient singularities. In my talk, I would like to give some related problems.