Seminarium Geometria Algebraiczna

This talk is based on the paper: G. G. Calbetó, On the finite irreducible subgroups of $GL_n(\mathbb C)$ and $PGL_n(\mathbb C)$ (arXiv:2510.00718)

Finite subgroups of $SL_n(\mathbb C)$ play important role in algebraic geometry, in particular they appear in the McKay Correspondence: if $G\subset SL_n(\mathbb C)$, $n=2,3$ is a finite subgroup, then the quotient $\mathbb C^n/G$ admits a crepant resolution of singularities. If $n=2$ then the crepant resolution is unique, if $n=3$ then there is a prefered crepant resolution of singularities. Moreover, the Euler characteristic of the crepant resolution equals the number of the conjugacy classes of the group $G$. The case of $n=2$ is classical, the first proof for $n=3$ was given by Roan by an explicit resolution for twelve types of groups.

I will start with a detailed discussion of the motivation, then explain results and main techniques.

Given a \(2n\)-dimensional smooth complex variety equipped with a nondegenerate holomorphic \(2\)-form, an \(n\)-dimensional subvariety is called Lagrangian if the form vanishes identically on it. This seminar is devoted to a class of surfaces, Lagrangian in their Albanese variety, obtained as the Galois closure of a rational map from a very general \((1,6)\) abelian surface to the projective plane. These surfaces share some other relevant properties and have interesting low invariants: \(K^2 = 24\), \(p_g=6\) and \(q=4\). The construction is governed by a unique (up to translation) genus \(6\) hyperelliptic curve in the linear system of the \((1,6)\) polarization. Part of the talk will concern the construction of these curves, which confirm a prediction of Bryan, Oberdieck, Pandharipande and Yin on the number of hyperelliptic curves in abelian surfaces.

In this talk I will briefly introduce the basic definitions for the Chow ring:

• algebraic cycles on schemes
• rational equivalence of algebraic cycles and the definition of the Chow group
• the moving lemma and Kleiman's transversality theorem
• the fundamental class of an algebraic scheme.

Chow ring is a basic object in the intersection theory and it can be considered as a variant of homology theory for quasi-projective varieties.

This talk is based on sections 1.2.1-1.3.2 of the book: D. Eisenbud, J. Harris: 3264 and All That, A Second Course in Algebraic Geometry.

I will discuss multiplicity of a scheme at a point in relation to intersection multiplicities. To this end, I will introduce the notions of the tangent cone as a generalisation of the tangent space to singular points of an algebraic variety, and of a blowing-up as a transform replacing a point of a variety with its projectivized tangent cone. Then I will define the first Chern class homomorphism and the canonical class of a variety, ending with the adjunction formula for the canonical bundle.

This talk is based on sections 1.3.8-1.4.3 of: D. Eisenbud, J. Harris: 3264 and All That, Intersection Theory in Algebraic Geometry.

This is the first of two lectures on the applications of algebraic geometry to tensors and algebraic combinatorics. In the first part of the talk, we will discuss the geometry and complexity of the matrix multiplication problem. Then we will focus on the notion of the generic subrank, following the work of Derksen, Makam and Zuiddam, as well as joint work with Paweł Pielasa and Matouš Šafránek.

This is the second lecture on the applications of algebraic geometry to tensors and algebraic combinatorics. We will begin by discussing one of the celebrated results by June Huh and his collaborators: the proof of the Heron–Rota–Welsh conjecture on the log-concavity of the absolute values of the coefficients of the characteristic polynomial of a (representable) matroid. We will then see how the geometric framework of this proof can be generalized, leading to interesting results on finite algebras. This is joint work with Jakub Jagiełła and Paweł Pielasa.

Polarized Hodge structures of the primitive cohomology of double EPW sextics are known to be isomorphic (up to Tate twist) to the vanishing cohomology of Gushel–Mukai fourfolds. On the other hand, together with Kapustka and Kapustka we showed that the primitive cohomology of double EPW cubes are isomorphic to the primitive cohomology of double EPW sextics. In the talk I will explain a direct argument proving that the primitive cohomology of double EPW cubes are isomorphic (up to Tate twist) to the vanishing cohomology of GM fourfolds, using a 20-nodal octic surface to relate them.

This is a joint work with Kuznetsov, Kapustka, and Kapustka.

We will compute the Chow rings of $\mathbb{P}^n$ and some related varieties, including Veronese varieties, duals of hypersurfaces, products of projective spaces, Segre varieties, graphs and blow-ups at a point. One of the tools used to the computations is a general form of Bezout's theorem. At the end of the talk we will use the Chow ring of blow-up at a point to prove that if $X,Y \subset \mathbb{P}^n$ are subschemes of complementary dimension and $p \in X \cap Y$, then under suitable conditions the intersection multiplicity $m_p(X,Y)$ equals $mult_p(X) mult_p(Y)$.

The talk is based on sections 2.1.1-2.1.10 of D. Eisenbud, J. Harris, 3264 and All That, Intersection Theory in Algebraic Geometry.

During the talk we apply previously obtained results involving the form of Chow rings of projective spaces and their products to gain understanding of certain moduli spaces. We will start by analyzing the space of all cubics on a projective plane which turns out to be another projective space. This insight will allow us to answer how many arbitrary cubic polynomials are needed to obtain a certain type of a singular curve as a linear combination of given equations. We will also quantify how many such linear combinations produce the desired singularity when we are given the right amount of cubics to interpolate between them.

In the second part of the talk we will take a look at a classical problem originally solved by Apollonius. Using appropriate moduli space we will show that given three general circles on a plane, there are exactly eight distinct circles which are tangent to those three given circles.

The talk is based on sections 2.2-2.5 of D. Eisenbud, J. Harris, 3264 and All That, Intersection Theory in Algebraic Geometry.

The geometry of irreducible holomorphic symplectic (IHS, sometimes referred to as hyperkähler) manifolds can be studied through the numerical properties of algebraic classes with respect to a non-degenerate quadratic form on the second cohomology group. In this context, a famous conjecture (SYZ) predicts that the existence of Lagrangian fibrations is detected by the presence of certain isotropic classes. While the conjecture holds in all known examples, it remains open in general. Recently, singular analogues of IHS manifolds have been proposed, providing a new framework to test the conjecture in a singular setting. In this talk, I will focus on Nikulin orbifolds, which are among the simplest singular examples, and present recent work classifying possible fibrations in this deformation class, from which the SYZ conjecture follows in this specific case.

Let $f(x)$, $g(x)$ be rational functions with complex coefficients. How does one determine the intersection of the subfields $C(f)$ and $C(g)$ inside $C(x)$? More generally, let $X \longleftarrow Y \longrightarrow W$ be branched covers between Riemann surfaces. Are there covering maps from $X$, $Y$, $W$ to a surface $P$, making the diagram commutative? If that is the case, we say that the correspondence fits in a Galois diagram. This seemingly elementary problem plays a central role in understanding how Jacobians decompose into simple components, up to isogeny. Suppose $Y \longrightarrow X$ is a ramified cover of algebraic curves such that there exists a curve $W$ for which $\operatorname{Jac}Y \sim \operatorname{Jac}X \times \operatorname{Jac}W$. A natural question is how to describe $W$. A guiding conjecture predicts that W can be identified using Galois-theoretic methods applied to the covering $Y \longrightarrow X$. It is equivalent to asking whether a certain correspondence $X \longleftarrow Y^* \longrightarrow W$ fits in a Galois diagram.

Equations of secant varieties play an important role in several branches of pure and applied mathematics. Landsberg and Manivel were among the pioneers in studying such equations using modern representation theory and algebraic geometry. Among the most useful equations are Koszul and Young flattenings, developed by Landsberg and Ottaviani, with ideas going back to Strassen. However, it was known that this approach in fact always produces equations for a larger cactus variety. I will report on recent joint work with Dolezalek in which we provide determinantal equations that distinguish secant varieties from cactus varieties. As an application, we obtain a simple, computer-free proof that the border rank of the $2\times2$ matrix multiplication tensor is seven.