Normal presentation on Elliptic Ruled surfaces

In this article we determine exactly which line bundles on elliptic ruled surface X are normally presented. In particular we see that numerical classes of normally presented divisors form a convex set. (recall that Num(X) is generated by the class of a minimal section C_0 and by the class of a fiber f and that C_0 is ample.) As a corollary of the above result we show that Mukai's conjecture is true for the normal presentation of the it 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) \geq 2g+2 on a curve X of genus g, then L embeds X with Koszul homogeneous coordinate ring.