Pure & Applied Algebraic Geometry
celebrating Giorgio Ottaviani's 60th birthday
An online event
June 21-25, 2021
Titles & abstracts
(You can download a .pdf file of schedule & abstracts here)
Hirotachi
Abo - The discriminant locus of a vector bundle VideoSlides
This talk is concerned with the classification of singular
elements of the space of global sections of a vector bundle on a
non-singular projective variety. A non-zero global section of a
vector bundle of rank r on an n-dimensional
non-singular projective variety defines a closed subscheme of
the projective variety called the zero scheme of the global
section. The zero scheme of a global section of a rank r
vector bundle has codimension at most r (if it is
non-empty). The global sections of the vector bundle whose zero
scheme is non-singular of codimension r form a Zariski
open set of the projective space of the global sections of the
vector bundle. We call the complement of such an open set the
discriminant locus of the vector bundle. The discriminant locus
of a vector bundle generally is irreducible of codimension one.
However, there are vector bundles whose discriminant loci are
not irreducible or have higher codimensions. Thus, it is a
tempting problem to classify such degenerate discriminant loci.
The main purpose of this talk is threefold; the first is to show
that the ampleness of the vector bundle on the non-singular
projective variety implies the irreducibility of its
discriminant locus, the second is to discuss the criteria that a
very ample vector bundle needs to satisfy, so that its
discriminant locus has codimension one, and the third is to use
these criteria to complete the classification of degenerate
discriminant loci for n=r.
This is joint work (work in progress) with Robert Lazarsfeld and
Gregory Smith.
Alberto Alzati - Some results about Weak Lefschetz
Property VideoSlides
The Weak Lefshetz Property deals with properties of
multiplication maps within quotient algebras A := C[
x_{0},...,x_{m} ]/I where I
is an Artinian homogeneous ideal generated by degree d
polynomials. WLP holds if the multiplication (between
homogeneous parts of A) by general linear element is of
maximal rank. Although the matter is rather elementary, there
are currently only a few results in this regard, lacking of a
general strategy, also for low values of m and d and for regular
sequences of r polynomials, for which it is conjectured that WLP
holds.
In the talk it will be proved that WLP holds when r = 5,
m = 4, d = 2; a case not previously considered.
Enrique Arrondo - Using algebraic geometry in the
representation theory of finite groups VideoSlides
Given a finite group G, the center of its group algebra
K(G) if a finite commutative K-algebra, so it
corresponds to a finite number of points. We use this idea to
reconstruct all the representation theory of G in terms
of those points. In particular, we will give a way of finding
all the irreducible representations of G and deciding
which fields K are suitable as ground fields for the
representations. We will show, as an application when G
is the symmetric group S_{d}, how to describe
the different symmetries of functions in d variables. As
a consequence, we will re-prove a theorem of Tocino stating that
the hyperdeterminant of a d-dimensional matrix is zero
for all but two types of symmetric matrices.
Alessandra Bernardi - Decomposition of polynomials and
minimal apolar schemes VideoSlides
I will present the relation among minimal apolar schemes of a
given form and its so-called Generalized Additive Decomposition.
Elisa Cazzador - Inverting catalecticants of ternary
quartics VideoSlides
We study the reciprocal variety to the LSSM of catalecticant
matrices associated with ternary quartics. With numerical tools,
we obtain 85 to be its degree and 36 to be the ML-degree of the
LSSM. We provide a geometric explanation to why equality between
these two invariants is not reached, as opposed to the case of
binary forms, by describing the intersection of the reciprocal
variety and the orthogonal of the LSSM in the rank loci.
Moreover, we prove that only the rank-1 locus, namely the
Veronese surface ν_{4}(P^{2}),
contributes to the degree of the reciprocal variety.
Luca Chiantini- Four times four VideoSlides
In the last years, methods for tensor analysis have involved a
series of increasingly deep tools for the study of projective
varieties. In return, the process suggested to algebraic
geometers a focus on some special aspects of the theory. I will
present an instance of this process, based on recent join works
with Ottaviani, Vanniuwenhoven, Bocci, Angelini, Mazzon, on the
construction of criteria for the minimality of Waring
decompositions of forms. The investigation is strictly linked to
the study of singularities of higher secant varieties. The
general ideas and methods will be illustrated with special
attention to the case of forms of degree four.
Ciro Ciliberto - Extensions of canonical curves and
double covers VideoSlides
A variety of dimension n is said to be extendable r
times if it is the linear section of a variety of dimension n+ r which is not a cone. I will recall some general facts
about extendability, with special regard for extensions of
canonical curves to K3 surfaces and Fano 3-folds. Then I
will focus on double covers and on their extendability
properties. In particular I will consider K3 surfaces of
genus 2, that are double covers of the plane branched over a
general sextic. A first results is that the general curve in the
linear system pull back of plane curves of degree k >
6 lies on a unique K3 surface, so it is only once
extendable. A second result is that, by contrast, if k
< 7 the general such curve is extendable to a higher
dimensional variety. In fact in the cases k = 4,5,6,
this gives the existence of singular index k Fano
varieties of dimensions 8, 5, 3, genera 17, 26, 37, and indices
6, 3, 1 respectively. For k = 6 one recovers the Fano
variety P(3, 1, 1,1), one of two Fano threefolds with
canonical Gorenstein singularities with the maximal genus 37,
found by Prokhorov. A further result is that this latter variety
is no further extendable. For k = 4 and 5 these Fano
varieties have been identified by Totaro.
Aldo Conca - Ideals associated to subspace arrangements Video
Given a subspace arrangement we may associate to it two ideals,
the intersection of the linear ideals associated to each
subspace or their product. The structure of the intersection
ideal is mostly unknown. For example already for a finite
collection of generic points in a projective space the degree of
the generators of the intersection is not known. On the other
hand the product ideal is better understood. An old theorem of
Herzog and myself asserts that the product ideal has a linear
resolution, or, which is the same, its regularity is given by
the number of subspaces in the arrandment. In the talk we will
discuss the structure of the resolution of the product ideal. We
will see that such a resolution is supported on a polymatroid.
In collaboration with Manolis Tsakiris of ShanghaiTech.
Jan Draisma - The geometry of GL-varieties VideoSlides
A GL-variety is an infinite-dimensional affine variety
equipped with a suitable action of the infinite-dimensional
general linear group. GL-varieties arise naturally in
the study of properties of polynomials (and more general
tensors) that do not depend on their number of variables, a
research theme that is attracting attention in diverse areas of
mathematics. I will report on joint work with Arthur Bik, Rob
Eggermont, and Andrew Snowden, on the structure of GL-varieties.
The
basic building blocks of GL-varieties are the affine
spaces A^{λ} corresponding to a finite tuple λ
of Schur functors. Indeed, one of our theorems says that any
irreducible GL-variety X admits a dominant
morphism B x A^{λ} →X for some λ
and some finite-dimensional variety B. If λ and B
are taken minimal, then λ is unique and the dimension of B
is the transcendence degree of the invariant function field K(X)^{GL}.
Maria Lucia Fania - Ulrich bundles on 3-dimensional
scrolls VideoSlides
I will report on a joint paper with M. Lelli-Chiesa and J. Pons
Llopis, where we construct Ulrich bundles of low rank on
3-dimensional scrolls, with a special attention to 3-folds in P^{5}
which are scrolls.
Maciej Gałązka - Distinguishing secant from cactus
varieties Video
Cactus varieties are defined using linear spans of arbitrary
finite schemes of bounded length, while secant varieties use
only isolated reduced points. In particular, any secant variety
is always contained in the respective cactus variety, and,
except in a few initial cases, the inclusion is strict. It is
known that lots of natural criteria that try to test membership
in secant varieties are actually only tests for membership in
cactus varieties. In this talk, we present the first techniques
to distinguish actual secant variety from the cactus variety in
the case of the Veronese variety. We focus on the case of κ_{14}(ν_{d}(P^{n})),
the simplest that exhibits the difference between cactus and
secant varieties. We show that for d > 4, the
component of the cactus variety κ_{14}(ν_{d}(P^{6}))
other than the secant variety consists of degree d
polynomials divisible by a (d−3)-th power of a linear
form. We generalize this description to an arbitrary number of
variables. We present an algorithm for deciding whether a point
in the cactus variety κ_{14}(ν_{d}(P^{n}))
belongs to the respective secant variety for d > 5, n
≠ 6. Our intermediate results give also a partial answer to
analogous problems for other cactus varieties to any Veronese
variety.
The talk is about a joint project with Tomasz Mandziuk and Filip
Rupniewski.
J.M. Landsberg - Tensors of minimal border rank VideoSlides
In his paper Symplectic bundles on the plane, secant
varieties and Lüroth quartics revisited GO gave a
determinantal description of Strassen's equations for secant
varieties of Segre varieties that led us to a vast
generalization of these equations (Young flattenings). There had
not been much progress regarding equations of secant varieties
after that until very recently we used the border apolarity of
Buczynska-Buczynski to obtain new equations.
I will report on work in progress with Chia-Yu Chang, Arpan Pal
and Joachim Jelisiejew on the geometry of these new equations
and how they compare to the old.
Laurent Manivel - Complete quadrics and Gaussian models Video
I will explain how the maximum likelihood degree (ML-degree) for
linear concentration models, as well as the algebraic degree of
semidefinite programming (SDP), can be understood in terms of
Schubert calculus on the very classical varieties of complete
quadrics. This allows to prove a conjecture by Sturmfels and
Uhler on the polynomiality of the ML-degree, and a conjecture by
Nie, Ranestad and Sturmfels providing an explicit formula for
the degree of SDP.
Joint work with M. Michalek, L. Monin, T. Seynnaeve and M.
Vodicka.
Emilia Mezzetti - Congruences of lines and families of
matrices of constant rank VideoSlides
I will report on recent joint work with A. Boralevi and M.L.
Fania, about quadric surfaces contained in the Pfaffian
hypersurface in P^{14}. I will then explain the
connections with congruences of lines in the 5-dimensional
projective space.
Rosa Maria Miró-Roig - The Tea Theorem VideoSlides
The goal of this talk is to show the ubiquity of the Weak
Lefschetz Property (WLP) and to prove the ”Tea Theorem”. More
precisely, I will establish a close relationship between a
priori two unrelated problems: (1) the existence of Togliatti
systems (i.e. homogeneous Artinian ideals I ⊂ k[x_{0},
· · · , x_{n}] generated by forms of degree d
which fail the WLP in degree d − 1; and (2) the
existence of (smooth) projective varieties X ⊂ P^{N}
satisfying at least one Laplace equation of order d − 1
≥ 2. These are two longstanding problems which lie at the
crossroads between Commutative Algebra, Algebraic Geometry,
Differential Geometry and Combinatorics. In the monomial case, I
will classify some relevant examples, I will establish minimal
and maximal bounds, depending on n and d ≥ 2,
for the number of generators of Togliatti systems. Finally, I
will related Galois coverings with cyclic group Z/d
to the so called GT-systems and study the minimality of
GT-systems in terms of the number of monomials that appear in
the expand of the determinant of a 3-line circulant matrix.
All I will say is based in joint work with either L. Colarte, P.
Di Poi, E. Evo, E. Mezzetti, M. Michałek G. Ottaviani, or M.
Salat.
Bernard Mourrain - Tensors, Eigenvectors and Simultaneous
DiaGOnalisation VideoSlides
Tensor decomposition is a problem which can be much difficult
than matrix rank decomposition, but it has many fascinating
facets both from the algebraic, geometric and application point
of view. In this talk, we will explore the relationship between
this difficult problem and standard linear algebra operations
such has eigenvector computation, simultaneous diagonalisation
of matrix pencils. We will describe some algebraic approaches
based on flat extension properties and simultaneous
diagonalisation for computing tensor decomposition. Some links
between varieties of commuting matrices and the Hilbert scheme
of points will be discussed.
Rita Pardini - Deformations of semi-smooth varieties
VideoSlides
A variety X is semi-smooth if locally in the étale
topology its singularities are either double crossing points (xy=0)
or pinch points (x^{2}-y^{2}z=0).
Alternatively, X is semi-smooth if it can be obtained
from a smooth variety X' by gluing it along a smooth
divisor D' via an involution g of D'. We
describe explicitly in terms of the triple (X', D', g)
the two sheaves on X that control its deformation
theory, that is, the tangent sheaf T_{X} and the
sheaf T^{1}_{X}:=ext^{1}(Ω_{X},O_{X}).
As an application, we discuss the smoothability of the
semi-smooth Godeaux surfaces (K^{2}=1, p_{g}=q=0).
This is joint work with Barbara Fantechi and Marco
Franciosi.
Frank-Olaf Schreyer - Godeaux surfaces VideoSlides
In this talk I report on joined work with Isabel Stenger. We
describe the construction of an 8-dimensional locally complete
family of simply connected numerical Godeaux surfaces, building
on an homological algebra approach. We also describe how the
families of Reid and Miyaoka with torsion Z/3Z
and Z/5Z arise in our homological setting.
Luca Sodomaco - Algebraic degree of optimization over a
variety with an application to p-norm distance degree VideoSlides
We study an optimization problem constrained on a real algebraic
variety X and whose parametric objective function f_{u}
is gradient solvable with respect to the parametric data u.
This class of problems includes Euclidean distance optimization
as well as maximum likelihood optimization. For these particular
optimizations, a prominent role is played by the ED and ML
correspondence, respectively. To our generalized optimization
problem, we attach an optimization correspondence and show that
it is equidimensional. This leads to the notion of algebraic
degree of optimization on X. We apply these results to p-norm
optimization, where p is a positive integer, and we
define the p-norm distance degree of X. When p=2,
we recover the ED degree introduced by Draisma, Horobeț,
Ottaviani, Sturmfels, and Thomas. Finally, we derive a formula
for the p-norm distance degree of X as a
weighted sum of the polar classes of X under suitable
transversality conditions.
This is joint work with Kaie Kubjas and Olga Kuznetsova.
Miruna-Ştefana Sorea - Signatures of paths and the
shuffle algebra Video
Our work is motivated by the theory of rough paths in stochastic
analysis, where information from a path is usually encoded in a
sequence of tensors with real entries, called the path
signature. Using tools from representation theory and applied
algebra, we prove an intriguing combinatorial identity in the
shuffle algebra. It has a close connection to de Bruijn’s
formula.
This talk is based on joint work with Laura Colmenarejo and
Joscha Diehl.
Bernd Sturmfels - Algebraic Statistics with a View
towards Physics VideoSlides
We discuss the algebraic geometry of maximum likelihood
estimation from the perspective of scattering amplitudes in
particle physics. A guiding example is the CHY model, where the
underlying very affine variety is the moduli space M_{0,n}
of n-pointed rational curves. The scattering potential
plays the role of the log-likelihood function, and its critical
points are solutions to rational function equations. Their
number coincides with the Euler characteristic. Soft limit
degenerations are combined with certified numerical methods for
concrete computations.
Jean Vallès - Free divisors in a pencil of plane
curves VideoSlides
Jerzy Weyman - Finite free resolutions and root
systems Video
I will discuss the connection between structure of perfect
ideals of codimension 3 and Gorenstein ideals of codimension 4
with root systems and Schubert varieties in homogeneous spaces.
This will be a friendlier version of discussion of this topic.