On base loci of higher fundamental forms of toric varieties

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Abstract

We study the base locus of the higher fundamental forms of a projective toric variety X at a general point. More precisely we consider the closure X of the image of a map (C)kPn, sending t to the vector of Laurent monomials with exponents p0,,pnZk. We prove that the m-th fundamental form of such an X at a general point has non empty base locus if and only if the points pi lie on a suitable degree-m affine hypersurface.

We then restrict to the case in which the points pi are all the lattice points of a lattice polytope and we give some applications of the above result. In particular we provide a classification for the second fundamental forms on toric surfaces, and we also give some new examples of weighted 3-dimensional projective spaces whose blowing up at a general point is not Mori dream.

Introduction

Let XPn be a projective variety and let qX be a general point. Denote by π:X˜X 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 XPn in order for the linear system |πHmE| to be special, which means that its dimension is bigger than the expected one. The problem has been widely studied in case m=2, 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 P2 (Segre-Harbourne-Gimigliano-Hirschowitz Conjecture [12], [16], [19], [27]) and P3 (Laface-Ugaglia Conjecture [21], [22]) at points in very general position. These conjectures predict that a necessary condition for |πHmE| 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 (C)kPn, so that X is a not necessarily normal toric variety. The principal tool that we use is the restricted linear system|πHmE|E, 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 |πHmE| too. On the other hand, the dimension of the m-th form turns out to be related to the speciality of the system |πH(m+1)E| (see Proposition 1.2).

In order to state our results, let us fix a k-dimensional lattice MZk and a finite set of points S={p0,,pn}M. It is possible to define a map f:(C)kPn which associates to t the vector of Laurent monomials with exponents p0,,pn. The closure of the image of the above map is a k-dimensional projective toric variety X(S)Pn, and we denote by 1X(S) the image of the neutral element of (C)k. The point 1 lies in the open torus orbit, and hence it is a general point of X(S). An element vN:=Hom(M,Z) defines a map CX by tf(tv), whose derivative at t=1 is a vector of T1X. This induces a linear map NZCT1X which allows us to identify P(NZC) with P(T1X)Pk1 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 p0,,pn 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 m2 the following are equivalent:

  • (1)

    the m-th fundamental form at 1X(S) has a base point [v]P(T1X);

  • (2)

    the points of S lie on an affine hypersurface of MZC of equation(vx)m+lower degree terms=0.

We then restrict to the case of a toric variety associated to a polytope. Indeed, given a full-dimensional lattice polytope ΔMZQ, it is possible to define a polarized pair (X,H), where X=X(Δ) is the projective toric variety associated to the lattice points ΔM, while H is a very ample divisor of X. In what follows we will denote by π:X˜X the blowing up of the toric variety X along the point 1 and by E the exceptional divisor. A first consequence of Theorem 1 is the following characterisation of projective toric surfaces whose second fundamental form at 1 is not full dimensional (see also Definition 1.8):

Proposition 2

Let ΔMZQQ2 be a full dimensional lattice polytope such that |ΔM|6 and let (X,H) 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 |πH3E| is special;

  • (3)

    Δ is either a Cayley polygon or it is equivalent, modulo GL(2,Z), to one of the following:

In particular the second fundamental form at 1 has non empty base locus if and only if Δ is Cayley.

Going back to the problem stated at the beginning of the introduction, an easy corollary of the above result is that if the linear system |πH3E| is special, then its base locus contains a curve (the strict transform of the closure of a one-parameter subgroup) intersecting E. We will show that if m4, this is no longer true, i.e. there are examples of special linear systems of the form |πHmE| whose base locus does not contain such a curve (see Example 3.4).

Finally, when k2, we make use of Theorem 1 in order to study stable base loci of divisors of the form πHmE on X˜. In particular we give a sufficient condition on Δ implying that πHmE is not semiample (Corollary 3.2) and as an application we provide the following new list of 3-dimensional weighted projective spaces P(a1,,a4), with ai30, whose blowing up at 1 is not a Mori dream space.

Proposition 3

Let X:=P(a1,,a4) and let H be an ample divisor of degree lcm(a1,,a4). If the vector of weights is in Table 1 then the divisor πHmE 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 ai belongs to the semigroup generated by the remaining ones. Concerning the list of [14], there is only one common case, namely P(17,18,20,27) (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 X(Δ) 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 XPn associated to a set S={p0,,pn} 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 S={p0,,pn}M of lattice points, and the map f:(C)kPn given by the Laurent monomials χp, for pS, whose image is the toric variety X=X(S) (see (1.3)). Given an integer m2, for simplicity of notation we will setJm:=Jm(f)|1andDr:=Dr(f)|1,

Applications

From now on we are going to apply the above results to projective toric varieties X:=X(Δ), associated to a full-dimensional lattice polytope ΔMZQ (see Subsection 1.3).

In particular, we first prove a corollary of Theorem 1, which allows to construct some interesting non semiample divisors on X˜, 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 ΔMQ and let us consider m:=lw(Δ)+1. Observe that if vN 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.

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