Claudia Stadlmayr at the ZAG Marathon

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