Normal presentation on elliptic ruled surfaces.

Abstract

From the introduction: Let X be an irreducible projective variety and L a very ample lLine bundle on X, whose complete linear series defines 'L : X ! P(H0(L)). Let S = 1
m=0 SmH0(X,L) and let R(L) = L1 n=0 H0(X,L n) be the homogeneous coordinate ring associated to L. Then R is a finitely generated graded module over S, so it has a
minimal graded free resolution. We say that the line bundle L is normally generated if the natural maps SmH0(X,L) ! H0(X,L m) are surjective for all m 2. If L is
normally generated, then we say that L satisfies property Np, if the matrices in the free resolution of R over S have linear entries until the p-th stage. In particular, property
N1 says that the homogeneous ideal I of X in P(H0(L)) is generated by quadrics. A line bundle satisfying property N1 is also called normally presented. Let R = kR1R2. . .
be a graded algebra over a field k. The algebra R is a Koszul ring iff TorRi (k, k) has pure degree i for all i. In this article we determine exactly (theorem 4.2) which line bundles on an elliptic ruled surface X are normally presented. As a corollary we show that Mukai’s conjecture is true for the normal presentation of the adjoint linear series for an elliptic ruled surface.
In section 5 of this article, we show that if L is normally presented on X then the homogeneous coordinate ring associated to L is Koszul. We also give a new proof of
the following result due to Butler: If deg(L) 2g + 2 on a curve X of genus g, then L embeds X with Koszul homogeneous coordinate ring.