On the Unirationality of del Pezzo surfaces of degree two

Abstract

Among geometrically rational surfaces, del Pezzo surfaces of degree two over a field k containing at least one point are arguably the simplest that are not known to be unirational over k. Looking for k-rational curves on these surfaces, we extend some earlier work of Manin on this subject. We then focus on the case where k is a finite field, where we show that all except possibly three explicit del Pezzo surfaces of degree two are unirational over k.

20 pages, Magma code included at the end of the source file

Cited by in corpus (8)

• The rationality problem for conic bundles
• The Clemens-Griffiths method over non-closed fields
• Del Pezzo surfaces over finite fields and their Frobenius traces
• Unirationality of del Pezzo surfaces of degree two over finite fields
• Unirationality of del Pezzo surfaces of degree two over finite fields (extended version)
• Geometry of the del Pezzo surface y^2=x^3+Am^6+Bn^6
• Density of rational points on a family of del Pezzo surfaces of degree one
• On the arithmetic of one del Pezzo surface over the field with three elements