Algebra & Geometry Seminar - Previous years: 2022-23
Algebra & Geometry Seminar - Previous years: 2022-23
Date: 21/09/2022 Time: 16:15 Room: 714
Title: The Kähler package for finite geometries and modular lattices
Abstract: In this joint work with Bill Huang, June Huh and Greg Smith we give very explicit proofs of the existence of a Kähler package for the graded Möbius algebra associated to the lattice of subspaces of a vector space over a finite field $\mathbb F_q$. There are fascinating connections to other areas such as the theory of Gelfand pairs and generalized Radon transforms.
Date: 28/09/2022 Time: 16:15 Room: 714
Title: A family of threefolds with canonical map of high degree
Abstract: In the first part of the talk I will review some known results on the degree of the canonical map of surfaces and threefolds of general type. I will then describe using an elementary approach the construction of a two-dimensional family of smooth minimal threefolds of general type with canonical map of degree 96, improving the previous known highest values of 72. If time allows I shall finally explain how we found the examples using the theory of abelian covers developed by Pardini and Liedtke. This talk is based on a joint work with Christian Gleißner.
Date: 06/10/2022 Time: 16:15 Room: 714
Title: Sagbi combinatorics of maximal minors
Abstract: By a theorem of Bernstein, Sturmfels and Zelevinsky the maximal minors of a generic matrix form a universal Gröbner basis (now a simple proof by Conca, De Negri and Gorla). The question whether the maximal minors are also a universal Sagbi basis for the corresponding Grassmannian was answered negatively by Speyer and Sturmfels who have a counterexample for 3 x 6 matrices and a lex monomial order. The joint project with Aldo Conca, on which we report, arose from the question whether the maximal minors are at least a revlex universal Sagbi basis. This is false as well, but needs a 3 x 8 matrix, whereas we can prove by exhaustion that 3 x 6 and 3 x 7 do not suffice for a counterexample.
In this context several other questions arise: are the initial algebras of the Grasmannian always normal? What about the initial algebra of the Rees algebra of the ideal of maximal minors? We will discuss our experimental findings. Another outcome of the project is a Hilbert series controlled Sagbi algorithm that uses Normaliz for monomial and binomial computations.
Date: 12/10/2022 Time: 16:15 Room: 714
Title: Chain algebras of finite distributive lattices
Abstract: This talk is based on my ongoing project with Professor Jürgen Herzog and Rodica Dinu. Let L be a finite distributive lattice and let t_1,...,t_n denote the elements of its ground set. To each maximal chain C of L one can associate a squarefree monomial m in K[t_1,...,t_n] which equals the product of all t_i belonging to C. We then consider the subalgebra K[m_1,...,m_s], generated by all such monomials, and call it the chain algebra of L.
In my talk I will discuss some properties of such algebras and their connection to combinatorial properties of the corresponding lattices. The main result of this talk will be an algebraic characterization of finite planar distributive lattices.
Date: 19/10/2022 Time: 16:15 Room: 714
Title: Hilbert-Kunz density function and its applications to some Hilbert-Kunz multiplicity conjectures
Abstract: In this talk we introduce a compactly supported and real valued continuous function called Hilbert-Kunz (HK) density function. We briefly describe its properties and its applications to study characteristic p-invariants like HK multiplicity and F-thresholds. Further we discuss in more detail some long standing conjectures of Watanabe-Yoshida and Yoshida on the HK multiplicities of quadric hypersurfaces. Here, using the classification of ACM bundles on the smooth quadric via matrix factorizations, we describe the HK density functions of the quadrics. The structure of the function explains the difficulties in nailing down the computations of HK multiplicities of even such simple class of rings. As a corollary we prove a part of the Watanabe-Yoshida conjecture for all dimensions. Moreover, for large p, we give a closed formula for HK multiplicities of quadrics hypersurfaces and a proof of Yoshida’s conjecture.
Date: 19/10/2022 Time: 17:30 Room: 714
Title: On finite presentation for the tame fundamental group
Abstract: This is a report on joint work with H. Esnault and M. Schusterman.
Recall that the etale fundamental group of a variety over an algebraically closed field of characteristic 0 is known to be a finitely presented profinite group; this is proved by first reducing to varieties over the complex numbers, and then comparing with the topological fundamental group. In positive characteristics, even if we restrict to smooth varieties, finite generation fails in general for etale fundamental groups of non-proper varieties (eg, for the affine line). For a smooth variety with a smooth, projective compactification with a SNC boundary divisor, we show that the tame fundamental group is a finitely presented profinite group. In particular, this holds for the fundamental groups of smooth projective varieties.
Date: 26/10/2022 Time: 16:15 Room: 714
Title: Rational angles among points in a plane lattice
Abstract: Given a plane lattice Λ we study the triples of elements A,B,C of Λ such that the angle ABC is a rational multiple of π. I will present a classification of the lattices and configurations which arise. This is a joint work with R. Dvornicich, D. Lombardo and U. Zannier
Date: 02/11/2022 Time: 16:15 Room: 714
Title: Wide coreflective subcategories
Abstract: A subcategory of a triangulated category is said to be localizing if it is closed under shifts, cones, and coproducts. The localizing subcategories of the derived category of a commutative noetherian ring were completely classifiied in work of Hopkins and Neeman; they are parametrized by subsets of the prime spectrum. But already for the simplest non-affine case, the derived category D(Qcoh P^1_k) of the category of quasicoherent sheaves on the projective line P^1 over an algebraically closed field k, the situation appears to be rather intricate. Krause and Stevenson have recently addressed the problem of classifying the strictly localizing subcategories of D(Qcoh P^1_k), that is, the localizing subcategories L appearing in semiorthogonal decompositions (L,M) of D(Qcoh P^1_k). Combining the classifications of smashing subcategories and tensor ideals, they obtain a class of strictly localizing subcategories of D(Qcoh P^1_k) which are parametrized by a copy of Z and the powerset of P^1, and they ask whether all strictly localizing subcategories arise in this way. We will see that, at least under certain set-theoretic assumptions, the answer is no: there exist “exotic” localizing subcategories beyond the ones constructed from our understanding of the compact objects.
Date: 16/11/2022 Time: 16:15 Room: 714
Title: Vector bundles and representations of hypersurface
Abstract: Let X be an integral hypersurface of degree d and dimension n in the projective space. The description of hypersurfaces as zero loci of suitable square matrices (possibly with some further properties) is a very classical topic in algebraic geometry. In this talk we show how the existence of particular classes of vector bundles supported on X yields the description of its defining equation as the degeneracy locus of a map between bundles. Then we show that every surface in the projective space of dimension three is the pfaffian of a map between Steiner bundles and finally we discuss some examples in higher dimensions. This is a joint work with Gianfranco Casnati.
Date: 23/11/2022 Time: 16:15 Room: 714
Title: Fano manifolds with Lefschetz defect 3
Abstract: The Lefschetz defect is a numerical invariant associated to a complex Fano manifold X, and it depends on the Picard number of prime divisors contained in it. Its main property is that if the Lefschetz defect of X is at least 4, then X is isomorphic to a product SxT, with S a del Pezzo surface (i.e. a Fano variety of dimension 2). In this talk we discuss a classification result for Fano manifolds with Lefshetz defect 3: although X is not necessarily a product, it still has a very explicit description. That is, there exist a smooth Fano variety T of dimension dim T = dim X - 2, and a fibration from X to T with fibres del Pezzo surfaces, such that the latter factorizes as a P^2-bundle over T and the blow-up along three smooth codimension 2 subvarieties. Finally, we see some applications of the structure theorem.
Date: 30/11/2022 Time: 16:15 Room: 714
Title: Results on the algebraic matroid of the determinantal variety
Abstract: This talk will report progress on the characterization of the algebraic matroid of the classical determinantal variety. A family of base sets of the matroid is presented and it is conjectured that it fully characterizes the matroid. The conjecture is reduced to a purely combinatorial statement, which is true for special cases. These results are enabled through the integration of a combinatorial notion of relaxed supports of linkage matching fields, an interpretation of the problem of bounded-rank matrix completion as a linear section problem on the Grassmannian, and a class of local coordinates on the Grassmannian described by Sturmfels and Zelevinsky in 1993.
Date: 14/12/2022 Time: 16:15 Room: 714
Title: Border Basis Schemes, new results and open problems
Abstract: Border Basis Schemes (BBS) have been intensively studied in the last years. The main reason is that they offer a computational approach to the study of those Hilbert Schemes which parametrize 0-dimensional schemes of a given multiplicity. In this talk some results and related open problems will be discussed. In particular, the problem of finding good embeddings of BBS will be examined, together with some questions related to their generic and special fibers.
Date: 21/12/2022 Time: 16:15 Room: 714
Title: K-polynomials of multiplicity-free varieties
Abstract: Multiplicity-free varieties are those whose mutidegrees are equal to zero or one. This special condition implies important arithmetic properties of the coordinate ring and its Gröbner degenerations. In this project, we show that the support of (twisted) K-polynomials of multiplicity-free varieties form a generalized polymatroid. We apply this to show that the Möbius support of a base linear polymatroid is a generalized polymatroid, and to settle a particular case of a conjecture of Monical, Tokcan, and Yong. This is joint work with Castilllo, Cid-Ruiz, and Mohammadi.
Date: 11/01/2023 Time: 16:15 Room: 713
Title: Hilbert schemes with few Borel-fixed points
Abstract: Given a Hilbert polynomial Q, one can construct a scheme, called the Hilbert scheme, that parameterizes subschemes in P^n with Hilbert polynomial Q. Every Hilbert scheme contains Borel-fixed points, and this was effectively used by Hartshorne to prove that the Hilbert schemes are connected. In this talk, I will describe Hilbert schemes with one or two Borel-fixed points and explain their geometry. Time permitting, I will suggest some future directions one can pursue.
Date: 18/01/2023 Time: 14:15 Room: 713
Title: Local positivity and effective Diophantine approximation
Abstract: In this talk I will discuss a new approach to prove effective results in Diophantine approximation relying on lower bounds of Seshadri constants. I will then show how to use it to prove an effective theorem on the simultaneous approximation of two algebraic numbers satisfying an algebraic equation.
Date: 25/01/2023 Time: 14:15 Room: 713
Title: Powers of monomial ideals with characteristic-dependence Betti numbers
Abstract: In this talk we deal with dependence of the Betti numbers of monomial ideals on the characteristic of the field. In our main results we provide constructions of monomial ideals such that all their powers have some characteristic-dependent Betti numbers. These tools allow to provide explicitly, for a fixed prime p, an edge ideal such that the Betti numbers of all its powers are different in characteristic 0 and characteristic p. Moreover we present a result about the so-called Kodiyalam polynomials.
This is a joint work with Antonio Macchia, Francesco Strazzanti and Volkmar Welker.
Date: 15/02/2023 Time: 16:15 Room: 509
Title: Perazzo algebras and the weak Lefschetz property
Abstract: A Perazzo polynomial is a form F of type F=p_0 x+p_1 y+ p_2 z+g, where p_0,p_1,p_2, and g are forms in the two variables u,v. These forms always have vanishing Hessian. They were first studied by Gordan and Noether in 1876, and then by Perazzo in 1900 in a geometric way. In the algebraic picture, we study "Perazzo algebras": a Perazzo algebra A_F is an artinian Gorenstein algebra with F as dual generator. The property of F having vanishing Hessian is translated to A_F as the failure of the strong Lefschetz property. This talk will present a study of the Perazzo algebras using both algebraic and geometric tools. The principal question is: "Does A_F fail the weak Lefschetz property or not?" . This is a joint work with N. Abdallah, N. Altafi, P. de Poi, A. Iarrobino, P. Macias Marques, E. Mezzetti, R. M. Miró-Roig, and L. Nicklasson.
Date: 22/02/2023 Time: 14:15 Room: 713
Title: Instanton sheaves of low charge on Fano threefolds
Abstract: Let X be a Fano threefold of Picard number one and of index 2 + h, h = 0, 1. An instanton sheaf of charge k on X is defined as a semi-stable rank 2 torsion free sheaf F with Chern classes c1 = −h, c2 = k, c3 = 0 and such that F (−1) has no cohomology. Locally free instantons, originally defined on the projective space and later generalised on other Fano threefolds X, had been largely studied from several authors in the past years; their moduli spaces present an extremely rich geometry and useful applications to thestudy of curves on X. In this talk I will illustrate several features of non-locally free instantons of low charge on 3 dimensional quadrics and cubics. I will focus in particular on the role that they play in the study of the Gieseker-Maruyama moduli space MX (2; −h, k, 0) and describe how we can still relate these sheaves to curves on X.
Date: 01/03/2023 Time: 14:15 Room: 713
Title: Rigid Hyperelliptic Fourfolds
Abstract: In this talk we provide a fine classification of rigid hyperelliptic manifolds in dimension four up to biholomorphism and diffeomorphism. These manifolds are described as finite étale quotients of a product of four Fermat elliptic curves. This is joint work with Andreas Demleitner.
Date: 08/03/2023 Time: 14:15 Room: 713
Title: On a new construction method for surfaces of general type with p_g=q=2
Abstract: In this talk we provide a new construction method for surfaces of general type with p_g=q=2. We give explicit and global equations for some known families of surfaces and also construct a new one. This is a joint work with Fabrizio Catanese.
Date: 15/03/2023 Time: 14:15 Room: 713
Title: Determinantal tensor product surfaces
Abstract: A tensor product surface S is an algebraic surface that is defined as the closure of the image of a rational map f from the product of two projective lines to the 3-dimensional projective space. In this talk, I will report on new determinantal representations of S under the assumptions that f is generically injective and its base points are finitely many and locally complete intersections. These determinantal representations are matrices that are built from the coefficients of linear relations (syzygies) and quadratic relations of the bihomogeneous polynomials defining f. This approach relies on a formalization and generalization of the method of "moving quadrics" introduced and studied by David Cox and his co-authors. This is joint work with Falai Chen (University of Science and Technology of China at Heifei).
Date: 15/03/2023 Time: 15:30 Room: 713
Title: Finite presentations of fundamental groups of moduli spaces
Abstract: In this talk braid group and their close cousins take a prominent role as fundamental groups in algebraic geometry. After discussing the moduli space of curves and elliptic curves in particular, this will also be demonstrated for new examples, moduli spaces of trigonal curves equipped with a canonical divisor contracted by the trigonal map.
Date: 22/03/2023 Time: 14:00 Room: 713
Title: On the number of rational points of curves over a surface in P3
Abstract: In this talk, we will show that the number of rational points of an irreducible curve of degree m defined over a finite field Fq lying on a surface S in P3 of degree d is, under certain conditions, bounded by m(d+q-1)/2. Within a certain range of m and q, this result improves all other known bounds in the context of space curves. The method we used is inspired by techniques developed by Stöhr and Voloch. In their seminal work of 1986, they introduced the Frobenius orders of a projective curve and used them to give an upper bound on the number of rational points of the curve. After recalling some general results on the theory of orders of a space curve, we will study the arithmetic properties of curves lying on a surface in P3, to finally prove the bound.
The talk is based on a joint work with J. Nardi, published at Acta Arithmetica.
Date: 22/03/2023 Time: 15:15 Room: 713
Title: Rank-Metric Codes: From Communication Networks to Semifields
Abstract: This talk offers an introduction to rank-metric codes and touches upon very recent developments in the research field. I will start with an overview on network coding, the context in which rank-metric codes find their main application. I will then illustrate some open problems about the mathematical structure of rank-metric codes and recent partial solutions, which link to fundamental questions in algebraic combinatorics and semifield theory.
Date: 05/04/2023 Time: 14:00 Room: 713
Title: On polynomial reducibility
Abstract: If A,B are subsets of a fixed set X, a reduction of A to B is a function f:X-->X --taken from a given class of functions-- such that A=f^{-1}(B). If the reduction f is reasonably tame, it witnesses that A is simpler than B, in the sense that the problem of membership in A is reduced to the same problem in B, via f.
Purpose of the talk is to give an introduction to some complexity hierarchies arising by considering specific classes of reductions; in particular, I discuss the relation of polynomial reducibility on subsets of a field k, presenting recent results, open questions, and directions for further research.
This is a joint work with C. Massaza.
Date: 12/04/2023 Time: 14:00 Room: 713
Title: Linkage and higher structure maps of perfect ideals in codimension 3
Abstract: Two classical problems in commutative algebra are the classification of the (perfect) ideals of a (regular local) ring in terms of their minimal free resolutions, and the description of their linkage classes. In particular one would like to describe all the ideals in the linkage class of a complete intersection. In this seminar we deal with these two problems for perfect ideals of codimension 3. The main tools that we use are sequences of linear maps that can be defined over an exact complex of length 3 and that generalize the well-known multiplicative structure (Joint work with Xianglong Ni and Jerzy Weyman).
Date: 19/04/2023 Time: 14:00 Room: 713
Title: Determinantal varieties from point configurations on hypersurfaces
Abstract: 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 projective space P^r 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 characterization of the singular locus of X_{2,d,n} and X_{3,2,n}. This is joint work with Alessio Caminata and Han-Bom Moon.
Date: 26/04/2023 Time: 14:00 Room: 713
Title: Double EPW's and irrational GM threefolds
Abstract: We construct two examples of projective IHS fourfolds of K3[2]-type with an action of the alternating group A7, making them some of the most symmetric IHS fourfolds. They are realized as so called double EPW-sextics and this allows us to construct an explicit family of irrational Gushel-Mukai threefolds.
Date: 03/05/2023 Time: 14:00 Room: 713
Title: Toric Sylvester forms
Abstract: We investigate the structure of the saturation of ideals generated by systems of sparse homogeneous polynomials over a toric variety with respect to the irrelevant ideal of its Cox ring. As our main results, we establish a duality property and make it explicit by introducing toric Sylvester forms, under a certain \sigma-positive assumption on the variety. In particular, we prove that toric Sylvester forms yield bases of some graded components of I^{sat}/I, where I denotes an ideal generated by n+1 generic forms, n is the dimension of the variety and I^{sat} is the saturation with respect to the irrelevant ideal. We illustrate the relevance of toric Sylvester forms by providing three consequences regarding elimination matrices, sparse resultants and toric residues. This is joint work with Laurent Buse (Université Côte d’Azur).
Date: 03/05/2023 Time: 15:00 Room: 713
Title: When does a perturbation of the equations preserve the normal cone?
Abstract: Let (R,m) be a local ring and J an arbitrary ideal of R. For any R-module M let \gr_J(M) := \bigoplus_{n \ge 0} J^nM/J^{n+1}M, which is the associated graded module of M with respect to J.
In a recent joint work, I and P.H. Quy proved the following theorem: Let I = (f_1,...,f_r), where f_1,...,f_r is a J-filter regular sequence. Then there exists a number N such that if f_i' = f_i mod J^N and I' = (f_1',...,f_r'), then gr_J(R/I) = gr_J(R/I'). If J is an m-primary ideal, this theorem implies a long standing conjecture of Srinivas and Trivedi on the invariance of Hilbert functions under small perturbations, which has been solved recently by Ma, Quy and Smirnov. As a byproduct of the proof, the Artin-Rees number of I and I' with respect to J are the same. Furthermore, we give explicit upper bounds for the smallest number N with the above property. These results solve two problems raised by Ma, Quy and Smirnov. We also prove a converse of the above theorem by showing that the condition I generated by a J-filter regular sequence is the best possible for its validity. The above theorem can be extended to perturbations with respect to filtrations of ideals. As a consequence, if R is a power series ring, f_1,...,f_r is a filter-regular sequence, and f_i' is the n-jet of f_i for n>>0, then I and I' have the same initial ideal with respect to any Noetherian monomial order. A special case of this consequence was a conjecture of Adamus and Seyedinejad on approximations of analytic complete intersection singularities.
Date: 10/05/2023 Time: 14:00 Room: 713
Title: Exceptional objects, spherical functors and lifting equivalences
Abstract: In a joint work with E. Macri, we study the relation between spherical and exceptional objects on categories D and E respectively, given by a Calabi-Yau spherical functor \Phi : D \to E. The aim of this construction is to provide a way to construct autoequivalences of a category E generated by exceptional objects. This relies on the Calabi-Yau spherical functor \Phi, and the autoequivalences of D that are 'compatible' with \Phi. The idea is an effort to generalize McMullen's construction of automorphisms of rational surfaces via marked cubics in P2. I will introduce all the notions involved, and give an idea of the examples one can hope to study with this construction.
Date: 17/05/2023 Time: 14:00 Room: 713
Title: A zero-sum condition for the flexibility of polyhedra
Abstract: A (triangulated) polyhedron flexes when we can realize it in the space in infinitely many non-isometric ways always with the same edge lengths. By introducing a suitable ambient space and using a degeneration technique, we show that a necessary condition for the flexibility of a polyhedron is that it admits a cycle the sum of whose edge lengths (possibly multiplied by -1) is zero.
This is a joint work with Georg Grasegger, Jan Legersky and Josef Schicho.
Date: 17/05/2023 Time: 15:15 Room: 713
Title: On the second fundamental form of cubic threefolds
Abstract: I will report on some geometric properties of the second fundamental form of the embedding which assigns to a cubic threefold X its intermediate Jacobian. The choice of a line on X defines a conic bundle structure on X over the projective plane. The discriminant curve is a quintic plane curve endowed with a natural etale double cover and the intermediate Jacobian is the Prym variety of this cover. We use the Hodge-Gaussian maps and the second Prym canonical Gaussian map associated with this quintic.
This is a joint work with E. Colombo, J.C. Naranjo and G.P. Pirola.
Date: 26/05/2023 Time: 16:00 Room: 715
Title: Algebraic Cryptoanalysis
Abstract: After introducing the main topics of cryptography and cryptoanalysis, we explain a universal method for converting cryptoanalytic attacks to the task of solving Boolean polynomial systems. Then we discuss several methods for solving such systems, for instance the Boolean Buchberger Algorithm and the application of SAT solvers.
A particularly powerful kind of attacks are fault attacks which we consider next. Using the block cipher LED-64 as an example, we study how to carry out such an attack in an actual case. The remainder of the talk is devoted to introducing a novel solving method for Boolean polynomial systems based on the sRes proof system. We compare this proof system to the previously considered ones and provide some sample applications.