Curvilinear base points, local complete intersection and Koszul syzygies in biprojective spaces

Abstract

Abstract. Let I = f 1 , f 2 , f 3 be a bigraded ideal in the bigraded polynomial ring k [s, u; t, v]. Assume that I has codimension 2. Then Z = V(I) ⊂ P 1 × P 1 is a finite set of points. We prove that if Z is a local complete intersection, then any syzygy of the f i vanishing at Z, and in a certain degree range, is in the module of Koszul syzygies. This is an analog of a recent result of Cox and Schenck (2003).

Degree and birationality of multi‐graded rational maps

“…However, there seems to be very few results and no general theory available for multi-graded rational maps, that is, maps that are defined by a collection of multi-homogeneous polynomials over a subvariety of a product of projective spaces. At the same time, there is an increasing interest in those maps, for both theoretic and applied purposes (see, for example, [4,5,34,52,53]). This paper is the first step toward the development of a general theory of rational and birational multi-graded maps, providing also new insights in the single-graded case.…”

Equations of parametric surfaces with base points via syzygies

“…The proofs of these results require an extension of the concept of regularity of a module, which is traditionally a concept for graded modules, to cover the case of bigraded modules. This extension was developed in a recent series of papers [9], [10]. We will start by summarizing the results needed from these papers, and prove some additional results needed for the application to our implicitization problem.…”