On the singularities of surfaces ruled by conics

TL;DR

This paper classifies the types of singularities on surfaces ruled by conics, showing they are rational double points of specific types, and describes how these surfaces relate birationally to minimal models through blow-ups.

Contribution

It provides a complete classification of singularities on conic-ruled surfaces and details their birational relationships with minimal models.

Findings

Singularities are rational double points of type A_n or D_n.

Singularities originate from specific blow-up sequences.

Characterization of minimal birational models of conic-ruled surfaces.

Abstract

We classify the singularities of a surface ruled by conics: they are rational double points of type $A_n$ or $D_n$. This is proved by showing that they arise from a precise series of blow-ups of a suitable surface geometrically ruled by conics. We determine also the family of such surfaces which are birational models of a given surface ruled by conics and obtained in a "minimal way" from it.