Higher Syzygies of Elliptic Ruled Surfaces

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Let L be a normally generated line bundle on X; we say L satisfies property Nₚ (notation after Mark Green) if the matrices in the free resolution of R (the homogeneous coordinate ring of X) over S (the homogeneous coordinate ring of the projective space corresponding to the complete linear series |L|) have linear entries until the p-th stage. In this article we prove the following result: Let X be an elliptic ruled surface and let L be a product of p+1 base point free and ample line bundles on X. Then L satisfies property Nₚ. In particular we prove that numerical classes of all divisors which satisfies property Nₚ form a convex set. (Recall that Num(X) is generated by the class of a minimal section C₀ and by the class of a fiber f and that C₀ is ample.) As a corollary of the above result we show that the adjoint bundle Kₓ+(2p+3)A satisfies property Nₚ, if A is an ample line bundle.