# Claudia Stadlmayr at the ZAG Marathon

## Weak del Pezzo surfaces with global vector fields

Our TopMath student Claudia Stadlmayr gave her first international conference talk as part of the "ZAG Marathon". The new online conference format took place on the occasion of the "Day of Knowledge" on 1 September 2020.

During the 24 hours of this day, 24 research lectures on algebraic geometry were given across all time zones - one of them by Claudia Stadlmayr on her research on "Weak del Pezzo surfaces with global vector fields".

**ZAG Seminar and ZAG Marathon**

*Text: Claudia Stadlmayr*

Over the past six months, the format of the ZAG seminar (short for "Zoom Algebraic Geometry Seminar"), an international research seminar on algebraic geometry that takes place online twice a week, has been established. With more than 40 organizers from all over the world, lectures on a wide variety of topics from algebraic geometry, and a lecture schedule that is already full until June 2021, it has developed into a very well-attended institution, helping the global algebraic geometry community to grow together in the past months whilst personal meetings and travel were not possible.

To celebrate the flourishing exchange of knowledge within the framework of this online format and science in general, Ivan Cheltsov, professor in Edinburgh and one of the organizers of the ZAG seminar, initiated a special edition of the ZAG seminar on the occasion of the Russian "Day of Knowledge" holiday on September 1st: "День Знаний": the "ZAG marathon". The "День Знаний" or "Day of Knowledge" is traditionally the date on which the school and university year begins in the Russian Federation and in many other former Soviet republics.

The idea of this unique marathon format was to celebrate each of the 24 hours of the "Day of Knowledge" with a research lecture on algebraic geometry from speakers located across all time zones. The marathon started in Sydney at 12 midnight (GMT), Claudia Stadlmayr represented our time zone at 12 noon (GMT) in Munich, and the closing lecture of the algebraic geometry marathon was held at 11pm (GMT) from Honululu.

The locations of the speakers are marked on the globe in the poster shown above. Video recordings of all talks can be found on the ZAG website.

**You can watch the talk on Vimeo: **

**Classification of weak del Pezzo surfaces with global vector fields**

*Text: Claudia Stadlmayr*

A classical result in algebraic geometry says that a non-degenerate surface in n-dimensional projective space is at least of degree (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 his work "Sulle superficie dell nmo ordine immerse nello spazio di n dimensioni" 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". A weaker form of the condition of the anti-canonical class being ample is to be only "big" and "nef", which intuitively means that the anti-canonical sheaf has "enough" global sections and "good" intersection behavior with curves. Such surfaces are called "weak del Pezzo surfaces" and can equivalently also be characterized by the fact that they are either the projective plane, the product of the projective line with itself, the second Hirzebruch surface or a blow-up of \(\mathbb{P}^2\) in at most eight points in almost general position.

Now, a natural question is what the symmetry groups of such surfaces are. Unfortunately, this question turns out to be difficult in general, since the automorphism group is a subgroup of the plane Cremona group of birational automorphisms of \(\mathbb{P}^2\), which is very large as well as still partly mysterious, and also the image of the embedding of Aut(X) in the Cremona group is difficult to determine. Therefore it is necessary to replace the question about the automorphism groups of such surfaces with a more accessible one, for which we observe the following:

According to a result by Alexander Grothendieck, the automorphism group is not just an abstract group, but the group of field-valued points of the automorphism scheme, whose connected component of the identity we would like to study.

The difficulties that arose in the investigation of the automorphism group can now be overcome when determining the connected automorphism schemes of weak del Pezzo surfaces with the help of "Blanchard's Lemma". This lemma says that for a birational morphism X \(\to\) Y between proper schemes over a field, the connected automorphism scheme of X embeds as a closed subscheme of the automorphism scheme of Y. Fortunately, the image of this closed embedding can be explicitly described in special situations: If X \(\to\) Y is the blow-up of Y in a closed point p of Y, then the connected automorphism scheme of X is the connected stabilizer subscheme of p.

The characterization of most of the weak del Pezzo surfaces as iterated blow-ups of the projective plane now makes it possible to determine their connected automorphism schemes by calculating the iterated stabilizer schemes of the blown up points.

The condition of being a blow-up of the projective plane in at most eight points in almost general position also puts restrictions on the types of curves that can occur on weak del Pezzo surfaces: As for curves with negative self-intersection, only (-1)-curves and (-2)-curves exist on weak del Pezzo surfaces.

Hence, we can now classify all weak del Pezzo surfaces with non-trivial connected automorphism schemes (or, equivalently, with global vector fields) and the configurations of (-1)- and (-2)-curves on them.

This was carried out in the article "Weak del Pezzo surfaces with global vector fields" over an algebraically closed field of arbitrary characteristic and thus answers a fundamental and very old question, looking back on the entire history of the study of del Pezzo surfaces, begun in 1887.

**Reference:**

- Weak del Pezzo surfaces with global vector fields (with G. Martin), 50 pages, 67 figures, arXiv:2007.03665

### Claudia Stadlmayr: Vita

- 2011-2012: Early Studies in philosophy at the University of Augsburg
- 10/2014-03/2018: Bachelor Mathematics at TUM
- 09/2016-03/2017: Study abroad at Université Pierre et Marie Curie (Paris VI), Sorbonne Universités, in "M2 de Mathématiques fondamentales"
- 04/2017-03/2018: TopMath Bachelor
- 2018: TopMath Study Award
- since 04/2018: TopMath Master (until 09/2020) and Doctorate studies
- 03/2020: Research stay at Steklov Mathematical Institute, Moscow, Russia
- Fellow of Bischöﬂiche Studienförderung Cusanuswerk e.V. (10/2014 to 09/2017), Max Weber-Programm Bayern (since 10/2014), and Studienstiftung des deutschen Volkes (01/2015-03/2020)

You can find more Information on the Homepage of Claudia Stadlmayr.

### Laudatio

"Already within her Bachelor thesis 'Deformations and Resolutions of Rational Double Points' Claudia Stadlmayr showed an extraordinarily deep understanding of algebraic geometry. She has to date written two preprints presenting her own results: 'Weak del Pezzo surfaces with global vector fields' (July 2020, 50 pages, together with Gebhard Martin) and 'Which rational double points occur on del Pezzo surfaces?' (September 2020, 20 pages), both of which have attracted the attention of experts.

(Weak) del Pezzo surfaces and, more generally, (weak) Fano varieties are central objects in algebraic geometry. They occur naturally in many classification lists and they are one of the building blocks in the so-called minimal model programme (also known as the Mori programme), which aims for nothing less than a complete classification of all algebraic varieties. The classification of the symmetry groups of these objects has a very long history, which includes the question about their vector fields. In dimension two, we now have a classification thanks to the work of Martin and Stadlmayr, which will surely find many applications.

In pure mathematics, this level of achievement at such an early career stage is highly remarkable - her first paper has already led to the invitation to the ZAG-Marathon in September and her second work is her Masters thesis." - Prof. Dr. Christian Liedtke (TUM), Mentor of Claudia Stadlmayr