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.

It is a second part of a talko 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) I will present a basic introduction to projective representation: definition, equivalence of projective representations and connections with Schur multipliers. Then I will recall definition of a primitive representation and present the role they play in the classification.

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 we discuss the results of joint work with Matteo Verzobio [1] about the common valuation of the division polynomials of points on elliptic curves. We prove a formula for the cancellation exponent between division polynomials psi and phi associated with a sequence of points on an elliptic curve defined over a discrete valuation field. The formula is identical with the result of Yabuta-Voutier [2] for the case of finite extension of $p$-adic field $\mathbb Q_p$ and generalizes to the case of non-standard Kodaira types for non-perfect residue fields. Our proof applied to the case of $\mathbb Q_p$ is much shorter and depends exclusively on the elementary properties of the Néron local heights.The study of the cancellation sequence has some interesting applications. For example, knowing the behaviour of it was necessary to show that every elliptic divisibility sequence satisfies a recurrence relation originally defined by Ward.

[1] B. Naskręcki, M. Verzobio, Common valuations of division polynomials, Proc. A. R. Soc. Edinb., 2024, DOI:10.1017/prm.2024.7, 1–15.
[2] P. Voutier and M. Yabuta, The greatest common valuation of $\phi_n$ and $\psi_n$ at points on elliptic curves, J. Number. Theory, 229 (2021), 16–38.

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.

In my talk I am going to elaborate on techniques for computing the Chow groups of several kinds of spaces:

• affine spaces,
• non-empty open subsets of affine spaces (I will introduce Mayer-Vietoris and excision theorems for that),
• schemes admitting an affine or quasi-affine stratification.

Then I am going to examine how Chow groups behave with respect to morphisms between varieties - for this I will need to formulate definitions of pushforward, pullback and generic transversality. In the end, I will introduce definitions of intersection multiplicity and dimensional transversality and show how much is generic transversality stronger than dimensional transversality.

This talk is based on sections 1.3.3-1.3.7 of the book: D. Eisenbud, J. Harris: 3264 and All That, Intersection Theory 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.

The physics of gauged linear sigma models offers a framework for constructing pairs of algebraic varieties that are expected to be derived equivalent and not birationally equivalent, often called double-mirrors. I will concentrate on a specific instance of this construction that produces a pair of double-mirror Calabi–Yau threefolds.

This is a joint work with Will Donovan, Wahei Hara, and Michał Kapustka.

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.

I will discuss a few unexpected applications of Galois theory to the arithmetic of division fields of abelian varieties. Except for some basic skills in algebra and geometry, no further knowledge of arithmetic geometry will be assumed.

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.