I will discuss the problem of comparing/computing the regularity of symbolic powers and regular powers of certain classes of squarefree monomial ideals focusing on edge ideals of graphs.
Herzog et al. characterized closed graphs as the graphs whose binomial edge ideals have a quadratic Groebner basis. In this talk, we focus on a generalization of closed graphs, namely weakly-closed graphs (or co-comparability graphs). Building on known results about Knutson ideals of generic matrices, we characterize weakly-closed graphs as the only graphs whose binomial edge ideals are Knutson ideals (associated with a certain polynomial f). In doing so, we re-prove Matsuda's theorem about the F-purity of binomial edge ideals of weakly-closed graphs in prime characteristic and we extend it to generalized binomial edge ideals.
Lastly, we will discuss some open conjectures on the F-purity of binomial edge ideals and on the relation between Knutson ideals and compatible ideals.
Let $k$ be a field and $R = k[x_1, \ldots, x_n]$. We obtain an improved upper bound for asymptotic resurgence of squarefree monomial ideals in $R$. We study the effect on the resurgence when sum, product and intersection of ideals are taken. We obtain sharp upper and lower bounds for the resurgence and asymptotic resurgence of cover ideals of finite simple graphs in terms of associated combinatorial invariants. We also explicitly compute the resurgence and asymptotic resurgence of cover ideals of several classes of graphs. We characterize a graph being bipartite in terms of the resurgence and asymptotic resurgence of edge and cover ideals. We also compute explicitly the resurgence and asymptotic resurgence of edge ideals of some classes of graphs.
Point configurations appear naturally in different contexts, ranging from the study of the geometry of data sets to questions in commutative algebra and algebraic geometry concerning determinantal varieties and invariant theory. In this talk, we bring these perspectives together: we consider the scheme X_{r,d,n} parametrizing n ordered points in r-dimensional projective space that lie on a common hypersurface of degree d. We show that this scheme has a determinantal structure and, if r>1, we prove that it is irreducible, Cohen-Macaulay, and normal. Moreover, we give an algebraic and geometric description of the singular locus of X_{r,d,n} in terms of Castelnuovo-Mumford regularity and d-normality. This yields a complete characterization of the singular locus of X_{2,d,n} and X_{3,2,n}. This is joint work with Han-Bom Moon and Luca Schaffler.
The celebrated results of Smith, Hara, and Mehta-Srinivas connect rational singularities in characteristic zero after reduction to characteristic p > 0 with F-rational singularities. In recent years, a number of invariants defined via Frobenius in positive characteristics have been introduced as quantitative measures of F-rationality. These include the F-rational signature (Hochster-Yao), relative F-rational signature (Smirnov-Tucker), and dual F-signature (Sannai). In this talk, I will discuss new results in joint work with Smirnov relating each of these invariants. In particular, we show that the relative F-rational signature and dual F-signature coincide, while also verifying that the dual F-signature limit converges.
10 February 2023, 6:30 pm IST (joining time 6:20 pm IST)
Saugata Basu, Purdue University, West Lafayette, IN, USA - Homology of symmetric semi-algebraic sets
Studying the homology groups of semi-algebraic subsets of $\mathbb{R}^n$ and obtaining upper boundson the Betti numbers has been a classical topic in real algebraic geometry beginning with the work of Petrovskii and Oleinik, Thom, and Milnor. In this talk I will consider semi-algebraic subsets of $\mathbb{R}^n$ which are defined by symmetric polynomials and are thus stable under the standard action of the symmetric group $\mathfrak{S}_n$ on $\mathbb{R}^n$. The homology groups (with rational coefficients) of such sets thus acquire extra structure as $\mathfrak{S}_n$-modules leading to possible refinements on the classical bounds. I will also mention some connections with a homological stability conjecture.
Joint work (separately) with Daniel Perrucci and Cordian Riener.
17 February 2023, 5:30 pm IST (joining time 5:20 pm IST)
Manuel Blickle, Johannes Gutenberg University Mainz, Rhineland Palatinate, Germany - The Canonical Dualizing Complex on F-finite Schemes
We show that on the category of F-finite Schemes with essentially finite type morphisms there is a dualizing complex $\omega^\bullet$ which is compatible with (-)^! pullback. In particular for the Frobenius morphism we have a canonical isomorphism $\omega^\bullet \cong F^!\omega^\bullet$. Up to now this used to be a standard assumption in the theory of $F$-singularities -- which as of now -- holds unconditionally. The approach follows an idea outlined by Ofer Gabber. All is joint work with Bhargav Bhatt, Karl Schwede and Kevin Tucker.
Let I be an ideal of a Noetherian local ring R. We study how properties of the ideal change under small perturbations, that is, when I is replaced by an ideal J which is the same as I modulo a large power of the maximal ideal. In particular, assuming that R/J has the same Hilbert function as R/I, we show that the Betti numbers of R/J coincide with those of R/I. We also compare the local cohomology modules of R/J with those of R/I.
We prove that the generic link of a generic determinantal ring of maximal minors is strongly F-regular, hence it has rational singularities. In the process, we strengthen a result of Chardin and Ulrich. They showed that the generic residual intersections of a complete intersection ring with rational singularities again have rational singularities. We show that they are, in fact, strongly F-regular.
In the mid 1990s, Hochster and Huneke showed that generic determinantal rings are strongly F-regular; however, their proof is quite involved. The techniques that we discuss will allow us to give a new and simple proof of the strong F-regularity of generic determinantal rings defined by maximal minors. Time permitting, we will also share a new proof of the strong F-regularity of determinantal rings defined by minors of any size. This is joint work with Yevgeniya Tarasova.
We introduce the notion of strongly Lech-independent ideals as a generalization of Lech-independent ideals defined by Lech and Hanes, and use this notion to derive inequalities on multiplicities of ideals. In particular we prove that if (R,m) and (S,n) are Noetherian local rings of the same dimension, S is a flat local extension of R,and up to completion S is standard graded over a field and I=mS is homogeneous, then the multiplicity of R is no greater than that of S.
Toric face rings, introduced by Stanley, are simultaneous generalizations of Stanley–Reisner rings and affine semigroup rings, among others. We use the combinatorics of the fan underlying these rings to inductively compute their rings of differential operators. Along the way, we discover a new differential characterization of the Gorenstein property for affine semigroup rings. Our approach applies to a more general class of rings, which we call algebras realized by retracts. This is joint work with C-Y. Chan, P. Klein, L. Matusevich, J. Page, and J. Vassilev.
Toric face rings, introduced by Stanley, are simultaneous generalizations of Stanley–Reisner rings and affine semigroup rings, among others. We use the combinatorics of the fan underlying these rings to inductively compute their rings of differential operators. Along the way, we discover a new differential characterization of the Gorenstein property for affine semigroup rings. Our approach applies to a more general class of rings, which we call algebras realized by retracts. This is joint work with C-Y. Chan, P. Klein, L. Matusevich, J. Page, and J. Vassilev.
Consider a homogeneous polynomial f in variables x_1,...,x_n. The set of polynomials obtained from f by permuting the variables in all possible ways generates an ideal, which we call a principal symmetric ideal. What can we say about the Betti numbers of a principal symmetric ideal? I will give a general answer in this talk. This is a joint work with Megumi Harada and Liana Sega.