impanga 2025/26

Although in recent years several authors have studied Cox rings of various varieties, such as log Fano varieties and moduli spaces of rational curves with marked points, little is known about Cox rings of varieties of general type. Consequently, only a few examples of Mori Dream spaces of general type are known. Moreover, in dimension two, a classification of Mori Dream surfaces of general type with geometric genus zero still seems far away.

A particular class of surfaces that appears promising to study is the so-called product-quotient surfaces. Furthermore, minimal product-quotient surfaces with geometric genus zero are already classified, so the main goal is to understand which among them are Mori Dream and which are not. In a recent paper, Keum and Lee showed, among other things, that product-quotient surfaces belonging to two specific families in the classification list are Mori Dream.

During the talk, we will briefly discuss the theory of product-quotient surfaces and explain the technique adopted by Keum and Lee to study their effective, nef, and semiample cones. We will then illustrate the main difficulties in studying the remaining product-quotient surfaces in the list and present some partial results in this direction, which are part of a joint work in progress with F. Polizzi.

I will address the question of adding closed points to a normal separated surface $X$, i.e. considering open embeddings of $X$ with complements of dimension zero. I will argue that to a surface $X$ one can add only finitely many points to get its saturated model, i.e. a surface to which one cannot add any points. I will show that the saturated model is unique and functorial in $X$. I will also describe how to construct it explicitly from the additive category of reflexive sheaves on $X$. I will say that a surface is saturated if it is isomorphic to its saturated model. I will discuss saturated surfaces and, more generally, saturated algebraic spaces of dimension two. The talk is based on joint work with A. Bondal, T. Pełka and D. Weissmann.