Computing syzygies over V[X1,…,Xk], V a valuation domain

Computing Tropical Varieties over Fields with Valuation

We show how tropical varieties of ideals I E K[x] over a field K with non-trivial valuation can always be traced back to tropical varieties of ideals π−1I E RJtK[x] over some dense subring R in its ring of integers. Moreover, for homogeneous ideals, we present algorithms on how the latter can be computed in finite time, provided that π−1I is generated by elements in R[t, x]. While doing so, we ...

Computing syzygies with Gröbner bases

The aim of this article is to motivate the inclusion of Gröbner bases in algebraic geometry via the computation of syzygies. In particular, we will discuss Gröbner bases for finite modules over polynomial rings, and mention how this tool can be used to compute minimal free resolutions of graded finite modules. We won’t prove any results. Instead, we refer the reader to the references at the end...

Dual Modules over a Valuation

On Higher Syzygies of Ruled Varieties over a Curve

For a vector bundle E of rank n + 1 over a smooth projective curve C of genus g, let X = PC(E) with projection map π : X → C. In this paper we investigate the minimal free resolution of homogeneous coordinate rings of X . We first clarify the relations between higher syzygies of very ample line bundles on X and higher syzygies of Veronese embedding of fibres of π by the same line bundle. More p...

On Syzygies of Ruled Varieties over a Curve

In this article we concern higher syzygies of line bundles on X = PC(E) where E is a vector bundle of rank n+1 over a smooth projective curve C of genus g. Let H be the tautological line bundle of X and projection π : X → C. Our main result is that for a = 1 or n = 1 or n = 2 and a = 2 (i.e. scrolls of arbitrary dimension or ruled surfaces or quadric surface fibrations), aH+π∗B satisfies Proper...