On the Singularities of Surfaces Ruled by Conics

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.

Key Words:

• Conic bundle
• Projective surface
• Rational singularities
• Ruled surface

2010 Mathematics Subject Classification:

• 14J26
• 14E05
• 14D06

ACKNOWLEDGMENTS

The referee of an earlier version of [3] pointed out that a certain argument we gave in that article was insufficiently clear. His observations led us to begin this further, more complete study: we are grateful to that referee for accuracy.

We are deeply indebted to Peter Newstead and M. S. Narasimhan for their warm support and helpful suggestions; we are also very grateful to Valentina Beorchia and Dario Portelli for many interesting discussions. This research has been supported by, funds of the Università degli Studi di Trieste–Finanziamento di Ateneo per progetti di ricerca scientifica–FRA 2011.

Notes

Communicated by R. Piene.