Geometric Syzygies of Elliptic Normal Curves and their Secant Varieties

We show that the linear syzygy spaces of elliptic normal curves, their secant varieties and of bielliptic canonical curves are spanned by geometric syzygies.     

Enriques surfaces and Jacobian elliptic K3 surfaces

This paper proposes a new geometric construction of Enriques surfaces. Its starting point are K3 surfaces with Jacobian elliptic fibration which arise from rational elliptic surfaces by a quadratic base change. The Enriques surfaces obtained in this way are characterised by elliptic fibrations with a rational curve as bisection which splits into two sections on the covering K3 surface. The construction has applications to the study of Enriques surfaces with specific automorphisms. It also allows us to answer a question of Beauville about Enriques surfaces whose Brauer groups show an exceptional behaviour. In a forthcoming paper, we will study arithmetic consequences of our construction.     

The Horrocks-Mumford bundle and the Ferrand construction

There is an analogue of the Ferrand construction for smooth surfaces in ₄: if the normal bundle of such a surface X has a suitable 1-subbundle, then a 2-vector bundle can be constructed, which has a section vanishing doubly on X. In this way the Horrocks-Mumford bundle is recovered from the quintic scroll.K. Stein, dem B ahnbrecher der Funktionen-theorie, gewidmet