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.