On base loci of higher fundamental forms of toric varieties☆
Introduction
Let be a projective variety and let be a general point. Denote by the blowing-up of X at q with exceptional divisor E. Given a hyperplane section H of X it is an open problem to provide necessary and sufficient conditions on the embedding in order for the linear system to be special, which means that its dimension is bigger than the expected one. The problem has been widely studied in case , see for instance [1], [6], [7] and the references therein, but it remains open even in this case. For higher values of m there are conjectures when X is the blowing up of (Segre-Harbourne-Gimigliano-Hirschowitz Conjecture [12], [16], [19], [27]) and (Laface-Ugaglia Conjecture [21], [22]) at points in very general position. These conjectures predict that a necessary condition for to be special is that it has positive dimensional base locus.
In this paper we investigate the above problem in case X is the closure of a monomial embedding , so that X is a not necessarily normal toric variety. The principal tool that we use is the restricted linear system which is also called the m-th fundamental form of X at q (see for instance [15], [20]). The m-th fundamental form is useful in two directions. On one hand, a base point for the m-th fundamental form is a base point for the system too. On the other hand, the dimension of the m-th form turns out to be related to the speciality of the system (see Proposition 1.2).
In order to state our results, let us fix a k-dimensional lattice and a finite set of points . It is possible to define a map which associates to t the vector of Laurent monomials with exponents . The closure of the image of the above map is a k-dimensional projective toric variety , and we denote by the image of the neutral element of . The point 1 lies in the open torus orbit, and hence it is a general point of . An element defines a map by , whose derivative at is a vector of . This induces a linear map which allows us to identify with in our main theorem. In [25] it is shown that the m-th fundamental form at 1 is not the complete linear system if and only if the points lie on an affine hypersurface of degree m. Our main result shows that the m-th fundamental form at 1 has a base point if and only if the top degree part of the above affine hypersurface is a pure power. More precisely we have the following (see also Example 2.3, Example 2.4 for the difference between our result and Perkinson's).
Theorem 1 Given an integer the following are equivalent: the m-th fundamental form at has a base point ; the points of S lie on an affine hypersurface of of equation
Proposition 2
Let be a full dimensional lattice polytope such that and let be the corresponding polarized pair. Then the following are equivalent:
- (1)
the second fundamental form of X at 1 is not full dimensional;
- (2)
the linear system is special;
- (3)
Δ is either a Cayley polygon or it is equivalent, modulo , to one of the following:
Finally, when , we make use of Theorem 1 in order to study stable base loci of divisors of the form on . In particular we give a sufficient condition on Δ implying that is not semiample (Corollary 3.2) and as an application we provide the following new list of 3-dimensional weighted projective spaces , with , whose blowing up at 1 is not a Mori dream space.
Proposition 3 Let and let H be an ample divisor of degree . If the vector of weights is in Table 1 then the divisor is nef but not semiample. In particular the blowing up of X at 1 is not Mori dream.
In [14] and [18] there are examples of 3-dimensional weighted projective spaces whose blowing up at 1 is not Mori dream. We remark that there is no intersection between our list and the one of [18], since we consider only the cases in which no weight belongs to the semigroup generated by the remaining ones. Concerning the list of [14], there is only one common case, namely (see also Remark 3.10).
The paper is structured as follows. In Section 1 we first introduce higher fundamental forms on projective varieties, then we recall some definitions and facts about projective toric varieties and finally we specialize to the case of toric varieties associated to a lattice polytope Δ. In Section 2 we prove Theorem 1 and we present a couple of related examples. The last section deals with some applications of Theorem 1 to toric varieties associated to a polytope Δ. In particular we first prove a corollary which gives a condition on Δ implying that a suitable divisor on the blowing up of the toric variety is not semiample. Then we consider the dimension 2 case, proving Proposition 2 and some related results, and finally we restrict to weighted projective spaces, proving Proposition 3.
We thank the anonymous referee whose comments helped us to improve the exposition of the paper.
Section snippets
Preliminaries
In this section we begin by recalling the definition of the m-th fundamental form of a projective variety, and then the definition of the projective toric variety associated to a set of lattice points. Finally we restrict to the case in which S is the set of all lattice points in a lattice polytope Δ.
Proof of Theorem 1
In this section we are going to prove Theorem 1, by following the idea of Perkinson [25], i.e. the study of suitable relations between left and right kernels of a slight modification of the m-th jet matrix.
Let us consider as in Subsection 1.2 any finite set of lattice points, and the map given by the Laurent monomials , for , whose image is the toric variety (see (1.3)). Given an integer , for simplicity of notation we will set
Applications
From now on we are going to apply the above results to projective toric varieties , associated to a full-dimensional lattice polytope (see Subsection 1.3).
In particular, we first prove a corollary of Theorem 1, which allows to construct some interesting non semiample divisors on , then we restrict to the case of toric surfaces and finally we consider weighted projective spaces.
Remark 3.1 Let us fix a lattice polytope and let us consider . Observe that if is a width
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The first author was partially supported by Proyecto FONDECYT Regular N. 119077. Both authors have been partially supported by project Anillo ACT 1415 PIA CONICYT.