Mapping toric varieties into low dimensional spaces

Abstract

Abstract. A smooth d-dimensional projective variety X can always be embedded into 2d + 1-dimensional space. In contrast, a singular variety may require an arbitrary large ambient space. If we relax our requirement and ask only that the map is injective, then any d-dimensional projective variety can be mapped injectively to 2d + 1-dimensional projective space. A natural question then arises: what is the minimal m such that a projective variety can be mapped injectively to m-dimensional projective space? In this paper we investigate this question for normal toric varieties, with our most complete results being for Segre-Veronese varieties. In particular, we find that in most cases the optimal m can be reached by performing linear projections from the defining embedding.

Derived Functors of Differential Operators

“…The radicial rank provides a lower bound for the dimension of the secant variety of this embedding: rra K (R) dim(Sec(X)) + 1. This is wellknown; see, e.g., [11,Corollary 4.3]. We note that this inequality can be sharp; see ibid.…”

“…Remark 5.3. We note that the local cohomology modules occurring in Proposition 4.1 are closely related to those that appear in the work [10,11] of Emilie Dufresne and the present author on invariant theory. A separating set for an action of a group G on a polynomial ring S = Sym(V * ) induced by a representation V of G consists of a set of invariants {f 1 , .…”

“…Computation of cohomological dimension is difficult in general, so direct application of Proposition 5.2 to study radicial rank and secant dimension is challenging. Examples to this effect appear in the work [10,11] discussed above. We give a couple more applications based on the following corollary.…”

“…This is applied to give meaningful lower bounds in terms of properties of the representation. A similar approach is executed in [11] to bound separating sets for actions of tori. It follows from Remark 2.4 that the bounds given by Proposition 5.2 are always at least as strong as those mentioned above; they also do not require the ring R to be realized as a subring of a polynomial ring.…”

Uniform symbolic topologies in normal toric rings

“…In passing, we note that the corresponding class of varieties (Segre-Veronese varieties) form a cornerstone of a lot of investigations in classical and applied algebraic geometry, often being varieties for which one has a more incisive handle on some specified computational-or other task. As a sampling of recent investigations in this vein, we include [2,3,4,9,22] among our references.…”