# TopMath Talk with Claudia Stadlmayr

## Del Pezzo surfaces and rational double points

On Wednesday, 4th November 2020 at 16:00, Claudia Stadlmayr from the Research group Algebra will give a talk on "Del Pezzo surfaces and rational double points" in a live stream. Everybody is welcome.

The live stream is done via Zoom: Please **contact the organizor fabian.roll (at) tum.de to get the login details**.

### Del Pezzo surfaces and rational double points

A classical result in algebraic geometry says that a non-degenerate surface in n-dimensional projective space is of degree at least (n-1). After Pasquale del Pezzo had classified such 'surfaces of minimal degree' in 1886, he studied algebraic surfaces of the next higher degree n in n-dimensional projective space in 1887. As an homage to del Pezzo's work, and slightly more generally, smooth projective surfaces with ample anti-canonical class have since been called 'del Pezzo surfaces'.

Omitting the smoothness condition, one can generalize this notion to the one of 'RDP del Pezzo surfaces' by allowing rational double points as singularities. Rational double points are classified by their dual resolution graphs, which are the Dynkin diagrams of types A, D, and E. In characteristic at least 7, rational double points are taut, that is, they are uniquely determined by their dual resolution graph, whereas in characteristic 2, 3, or 5 there sometimes are more rational double points belonging to the same Dynkin diagram, and these have been destinguished by Artin in terms of their deformation theory.

In 1934, Du Val classified all configurations of rational double points that can appear on complex RDP del Pezzo surfaces. Due to the existence of non-taut rational double points, this question becomes very subtle in positive and small characteristics. In this talk, I will explain the above notions and finish with an outline of my recent work, which answers the question 'Which rational double points occur on del Pezzo surfaces?' in all characteristics.

### TopMath Talks

As part of the TopMath talks, TopMath students and doctoral students present parts of their research. They provide an understandable insight into their current area of interest, enabling students and staff to broaden their mathematical background knowledge. The talks are open to the public and last about an hour, followed by discussion. You are cordially invited!