Automorphisms of real rational surfaces and weighted blow-up singularities

TL;DR

This paper studies the automorphism groups of singular real rational surfaces created via weighted blow-ups, demonstrating their transitive action on certain point configurations and classifying these surfaces.

Contribution

It generalizes previous results by removing nonsingularity assumptions and provides a classification of singular real rational surfaces from weighted blow-ups.

Findings

Aut(X) acts transitively on X^e for weighted blow-up points.

Generalizes earlier work to singular surfaces.

Classifies singular real rational surfaces obtained from nonsingular ones.

Abstract

Let X be a singular real rational surface obtained from a smooth real rational surface by performing weighted blow-ups. Denote by Aut(X) the group of algebraic automorphisms of X into itself. Let n be a natural integer and let e=[e_1,...,e_l] be a partition of n. Denote by X^e the set of l-tuples (P_1,...,P_l) of distinct nonsingular curvilinear infinitely near points of X of orders (e_1,...,e_l). We show that the group Aut(X) acts transitively on X^e. This statement generalizes earlier work where the case of the trivial partition e=[1,...,1] was treated under the supplementary condition that X is nonsingular. As an application we classify singular real rational surfaces obtained from nonsingular surfaces by performing weighted blow-ups.