# Mixed Threefolds Isogenous to a Product

## TL;DR

This paper classifies threefolds isogenous to a product of mixed type with a specific Euler characteristic, using computational group theory techniques to systematically construct and analyze these complex algebraic varieties.

## Contribution

It provides the first complete classification of certain threefolds isogenous to a product with negative Euler characteristic, employing a MAGMA algorithm for systematic analysis.

## Key findings

- Classified all threefolds with (\u2113}_X)=-1 under specified automorphism conditions.
- Developed a MAGMA algorithm for constructing these threefolds.
- Identified these varieties as boundary cases in threefold geography.

## Abstract

In this paper we study \emph{threefolds isogenous to a product of mixed type} i.e. quotients of a product of three compact Riemann surfaces $C_i$ of genus at least two by the action of a finite group $G$, which is free, but not diagonal. In particular, we are interested in the systematic construction and classification of these varieties. Our main result is the full classification of threefolds isogenous to a product of mixed type with $\chi(\mathcal O_X)=-1$ assuming that any automorphism in $G$, which restricts to the trivial element in $Aut(C_i)$ for some $C_i$, is the identity on the product. Since the holomorphic Euler-Poincar\'e-characteristic of a smooth threefold of general type with ample canonical class is always negative, these examples lie on the boundary, in the sense of threefold geography. To achieve our result we use techniques from computational group theory. Indeed, we develop a MAGMA algorithm to classify these threefolds for any given value of $\chi(\mathcal O_X)$.