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.