Talks
- Mariemi Alonso (Universidad Complutense de Madrid):
Some "effective-results" in the theory of Henselian Rings
In this talk we present some results about general henselian rings; namely Local Bezout theorem and a generalization of Weierstrass division theorem (A.-H.Lombardi Collect. Math 2016). Our approach is constructive in the sense of Bishop and previous works of the authors; so that they have a strong algorithmic character. In the geometric frame the effectiveness of henselian division was achieved in (A.-Hauser- Castro, FoCM 2018). Both in the geometric and abstract cases Multivariate Hensel Lemma is a main tool. We discuss the algorithmic behaviour of it; that is, the reduction from Multivariate to Univariate Hensel Lemma, in different contexts eg. local rings, henselian valuations rings, that in the Literature are often deduced for local rings using a version of ZMT "a la Peskine", whose algorithmic approach is harder (cf.A.-H.Lombardy-T.Coquand, J. of Algebra 2014).The results of this talk have been obtained or are a work in progress in collaboration with H. Lombardi. - Josep Àlvarez Montaner (Universitat Politècnica de Catalunya):
Poincaré series of multiplier and test ideals
We prove the rationality of the Poincaré series of multiplier ideals in any dimension and thus extending the main results for surfaces of Galindo and Monserrat and Alberich-Carramiñana et al. Our results also hold for Poincaré series of test ideals. In order to do so, we introduce a theory of Hilbert functions indexed over the real numbers which gives an unified treatment of both cases. Joint work with Luis Núñez-Betancourt. - Ana Bravo (Universidad Autónoma de Madrid):
A singular journey
We will review the role of the order function on constructive resolution of singularities. Joint with A. Benito and S. Encinas. - Alessio Caminata (Università di Genova):
Determinantal varieties arising from point configurations on hypersurfaces
We consider the scheme X_{r,d,n} parametrizing n ordered points in projective space P^r that lie on a common hypersurface of degree d. We show that this scheme has a determinantal structure and we prove that it is irreducible, arithmetically 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. For points on plane curves, we show that this description is related to some classical theorems about algebraic curves. This is a preliminary report on joint work with Han-Bom Moon and Luca Schaffler. - Alberto Castaño Domínguez (Universidad de Sevilla):
Hodge ideals of some free divisors
Hodge ideals are a tool introduced by Popa and Mustaţă to study birrationally the Hodge filtration on the sheaf of meromorphic functions on a variety with poles along certain divisor. The first of these ideal sheaves is a multiplier ideal, so the former form a finer invariant than the latter in the study of singularities.
However, Hodge ideals are usually hard to determine, especially in the case of non-isolated singularities, as free divisors have. For some of them, certain properties of the roots of their associated Bernstein-Sato polynomials, together with some results from Hodge module theory, allow us to give a purely algebraic description of their Hodge ideals. In this talk I will comment on these results and examples, which are a joint work with Christian Sevenheck and Luis Narváez Macarro. - Yairon Cid Ruiz (Universiteit Gent):
The fiber-full scheme
We introduce the fiber-full scheme which can be seen as the parameter space that generalizes the Hilbert and Quot schemes by controlling the entire cohomological data. In other words, the fiber-full scheme controls the dimension of all cohomologies of all possible twistings, instead of just the Hilbert polynomial. We also present some applications that can be derived from the existence of the fiber-full scheme. This talk is based on joint work with Ritvik Ramkumar. - Teresa Cortadellas (Universitat de Barcelona):
Blowup algebras
Blowup algebras appear in many constructions in commutative algebra and algebraic geometry.
In this talk I will expose some of the contributions about blowup algebras made with Santiago Zarzuela. These results focus on the study of the relationships between the Rees algebra, the associated graded ring and the fiber cone of a filtration of ideals of a local ring; and also of the situations that assure good properties, as the Cohen-Macaulay property, of these rings. - Celia del Buey (Universidad Autónoma de Madrid):
Yuan’s correspondence for Galois ring extensions of exponent one
Yuan presents an exponent one inseparable Galois theory for commutative rings extensions of prime characteristic p >0. We say that an extension of exponent one, A⊂ C, is a Galois extension if C is finitely generated projective as A-module and locally has p-basis. Yuan proves that this local condition is equivalent to the hypothesis C[Der_A(C)]=Hom_A(C,C). Using this characterization, he establishes a correspondence between the intermediate rings between C and A over which locally C admits p-basis, and the restricted Lie subalgebras of Der_A(C) which are also C-module direct summands of Der_A(C). We will review the concepts and the mains ideas of Yuan’s work as an introduction to our results on the representability of Yuan’s functor, that will be exposed later by Diego Sulca. The results on representability are joint work with Diego Sulca (FAMAF, Universidad Nacional de Córdoba, Argentina) and Orlando Villamayor (Universidad Autónoma de Madrid). - Eleonore Faber (University of Leeds):
Discriminants of reflection groups and noncommutative resolutions
Let G be a finite subgroup of GL(n,K) for a field K whose characteristic does not divide the order of G. The group G acts linearly on the polynomial ring S in n variables over K. When G is generated by reflections, then the discriminant D of the group action of G on S is a hypersurface with a singular locus of codimension 1, in particular, D is a free divisor.
In this talk we explain a natural construction of a "noncommutative" resolution of singularities of the coordinate ring of D. This yields a McKay correspondence for reflection groups. Further we are interested in determining the "components" of this noncommutative resolution, that are certain maximal Cohen-Macaulay modules over the coordinate ring of D. This is joint work with Ragnar Buchweitz, and Colin Ingalls, and work in progress with Ingalls, Simon May, and Marco Talarico. - Herwig Hauser (Universität Wien):
The phylogenetic tree of n points on the projective line
The Deligne-Mumford-Knudsen compactification of the moduli space M_{0,n} of n different points on the projective line proposes a precise notion of limit as some of the points come together and coalesce. This is encapsulated in the notion of n-pointed stable curves. Proofs require quite a bit - if not a lot - of machinery from algebraic geometry and commutative algebra.
In the talk, these limits of n different points are defined in an alternative and rather elementary way through a suitable and very natural embedding of M_{0,n} into a huge projective space: it then suffices to just take the Zariski-closure of the image to get the limits.
The proofs are then done through the use of phylogenetic trees - a graph-theoretic bookkeeping of how the different points come together. This turns out to be miraculously practical: for most of the proofs, one just has to look at the phylogenetic tree and copy the recipe "printed there" of how to proceed.
Collaboration with Jiayue Qi and Josef Schicho from the University of Linz. - Abbas Nasrollah Nejad (Institute for Advances Studies in Basic Sciences):
The Jacobian relation type of hypersurface singularities
In this talk, we introduce the notion of the Jacobian relation type of affine, analytic and formal algebras over a field and show that it is well-defined and invariant. In particular, the Jacobian relation type is an invariant of algebraic, analytic, and algebroid varieties. Then we will discuss and assay the Jacobian relation type of hypersurface singularities. This talk is based on joint ongoing work with Maryam Akhavin. - Ana J. Reguera (Universidad de Valladolid):
Small irreducible components of arc spaces in positive characteristic
Joint work with A. Benito and O. Piltant.
In 1968, J. Nash initiated the study of the space of arcs X_∞ of a (singular) algebraic variety X over a field of characteristic zero, with the purpose of understanding the structure of the various resolutions of singularities of X. His work was done shortly after Hironaka’s proof of Resolution of Singularities in characteristic zero. Nash proved, using Resolution of Singularities that the space of arcs X^{Sing}_{∞} centered in the singular locus of X has a finite number of irreducible components.
This Nash program extends, with some important differences, to perfect ground fields k of characteristic p > 0. The first difference is that Resolution of Singularities is still an open problem if char k = p > 0 and dim X ≥ 4. Another difference is that, in contrast with characteristic zero, (Sing X)_∞ may contain some of the irreducible components of X^{Sing}_{∞}. Understanding these “small” components is the main purpose of our article [1].
In this talk, we will propose some questions which would have an affirmative answer if a resolution of singularities existed:
Q1: Has X^{Sing}_{∞} a finite number of irreducible components?
Q2: Given a variety X, does there exist a proper and birational morphism Y ⟶ X such that Y_∞ is irreducible?
We will give partial answers and explain the status of these problems.
[1] A. Benito, O. Piltant, A.J. Reguera, Small irreducible components of arc spaces in positive characteristic, J. Pure Appl. Algebra 226 (2022). - Francesco Strazzanti (Università di Torino):
The Hilbert-Kunz function of some quadratic quotients of the Rees algebra
Given a commutative ring R and an ideal I of R, a family of rings obtained as suitable quotients of the Rees algebra of I has been studied as a unified approach to existing constructions and as a way to produce new interesting examples, especially integral domains. After reviewing its main properties I will focus on the Hilbert-Kunz functions and Hilbert-Kunz multiplicities of the rings in this family.
This is joint work with Santiago Zarzuela. - Diego Sulca (Universidad Nacional de Córdoba):
The Grassmannian of the Galois subextensions of a Galois extension of exponent one
This is a continuation of the talk given by Celia del Buey. Let A⊂ C be a Galois extension of exponent one and rank p^n. Given 0<r<n, we present an A-scheme Y_r that parameterizes the intermediate extensions A⊆ B⊆ C such that B ⊆ C is Galois of rank p^r. We show that this scheme is quasi-projective over A. Indeed, we construct Y_r as a closed subscheme of the restriction of scalars Res_{C/A} G_{n,r}, where G_{n.r} is the usual Grassmannian over C defined from the locally free C-module Der_A(C) (of rank n). We finally show that Y_r is smooth when A is a field and compute its dimension. This is joint work with Celia del Buey and Orlando Villamayor (Universidad Autónoma de Madrid). - Mari Paz Tirado Hernández (Universidad Autónoma de Madrid):
Hasse-Schmidt derivations, integrability and leaps
We will review the main properties of Hasse-Schmidt derivations and integrability, focusing on the case in which the base ring has positive characteristic, and some results about the set of leaps (in the sense of Hasse-Schmidt) will be presented.
