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Local positivity of line bundles on smooth toric varieties and Cayley polytopes. (English) Zbl 1329.14023

In the present paper the author studies the positivity phenomenon for toric varieties using the theory of Cayley polytopes. Let us recall basic notions, which we need to formulate main results of the paper.
Let \(P_{0},\dots, P{r} \subset \mathbb{R}^{p}\) be polytopes. One defines \[ [P_{0} \star \dots \star P{r}]^{k} = \text{Conv}\{(P_{0} \times 0) \cup (P_{1} \times ke_{1}) \cup \dots \cup (P_{r} \times ke_{r}) \} \subset \mathbb{R}^{p} \times \mathbb{R}^{r}, \] where \(e_{1}, \dots, e_{r}\) is the canonical basis for \(\mathbb{R}^{r}\). A polytope \(P \subset \mathbb{R}^{n}\) is called a Cayley polytope of order \(k\) and length \(r+1\) if there exist some lower dimensional polytopes \(P_{0}, \dots, P_{r}\) such that \(P\cong [P_{0} \star \dots \star P_{r}]^{k}\). If \(P_{0}, \dots, P_{r}\) can be taken to be normally equivalent, i.e. to have the same normal fan \(\Sigma\), then \(P\) is called a strict Cayley polytope and one writes \(\text{Cayley}^{k}_{\Sigma}(P_{0}, \dots, P_{r})\) for \([P_{0} \star \dots \star P_{r}]^{k}\).
For a line bundle on a smooth variety \(X\) and a point \(x \in X\) with the maximal ideal \(\mathfrak{m}_{x} \subseteq \mathcal{O}_{X}\) consider the map
\[ j_{x}^{k} : H^{0}(X, \mathcal{L}) \rightarrow H^{0}(X, \mathcal{L} \otimes (\mathcal{O}_{X} / \mathfrak{m}_{x}^{k+1})). \]
The projective linear subspace \(\mathbb{T}_{x}^{k}(X, \mathcal{L}):= \mathbb{P}(Im(j_{x}^{k}))\) of \(\mathbb{P}(H^{0}(X, \mathcal{L} \otimes (\mathcal{O}_{X} / \mathfrak{m}_{x}^{k+1})))\) is called the osculating space of order \(k\) at \(x \in X\). When the map \(j_{x}^{k}\) is onto, then one says that \(\mathcal{L}\) is \(k\)-jet spanned at \(x \in X\). Moreover, if \(\mathcal{L}\) is \(k\)-jet spanned at every point, one says that \(\mathcal{L}\) is \(k\)-jet spanned on \(X\). Finally, the largest \(k\) such that \(X\) is \(k\)-jet spanned at \(x \in X\) is denoted by \(s(\mathcal{L},x)\).
Moreover, for a nef line bundle \(\mathcal{L}\) on a smooth projective variety \(X\) the Seshadri constant of \(\mathcal{L}\) at \(x \in X\) is defined as \[ \varepsilon(X, \mathcal{L};x) := \text{inf}_{C \subseteq X} \frac{\mathcal{L}.C}{m_{x}(C)}, \] where the infimum is taken over all irreducible curves \(C\) passing through \(x\) and \(m_{x}(C)\) denotes the multiplicity of \(C\) at \(x\).
The first result of this note shows that the polytope associated to a smooth polarized toric variety with certain prescribed local positivity properties has a Cayley structure and vice versa.
{Theorem 1.} Let \((X, \mathcal{L})\) be a smooth polarized toric variety and let \(P_{\mathcal{L}}\) be the polytope associated to the complete linear series \(|\mathcal{L}|\). Then for any fixed \(k \in \mathbb{N}\) we have that \(s(\mathcal{L},x) =k\) for all points \(x \in X\) if and only if \(P \cong [P_{0} \star P_{1}]^{k}\) for some lower dimensional polytopes \(P_{0}\) and \(P_{1}\) and every edge of \(P\) has lattice length at least \(k\).
The following result gives a link between Seshadri constants, \(s\)-invariants and Cayley polytopes.
{Theorem 2.} Let \((X, \mathcal{L})\) be a smooth polarized toric variety, \(P_{\mathcal{L}}\) the corresponding smooth polytope and pick \(k \in \mathbb{N}\). Then the following conditions are equivalent:
i)
\(s(\mathcal{L},x) = k\) at every point \(x \in X\),
ii)
\(s(\mathcal{L},x) = k\) at fixed points and at a general point,
iii)
\(\varepsilon(X, \mathcal{L};x) = k\) at every point \(x \in X\),
iv)
\(\varepsilon(X, \mathcal{L};x) = k\) at fixed points and a general point,
v)
\(P_{\mathcal{L}} \cong [P_{0} \star P_{1}]^{k}\) for some lower dimensional polytopes \(P_{0}\) and \(P_{1}\) and every edge of \(P\) has lattice length at least \(k\).
The last result of the note shows that Theorem 1 is a generalization of the classification given by D. Perkinson [Mich. Math. J. 48, 483–515 (2000; Zbl 1085.14516)].
{Theorem 3.} Let \(P\) be a smooth polytope of dimension at most \(3\). If \(P \cong [P_{0} \star P_{1}]^{k}\) for some \(k \in \mathbb{N}\) and lower dimensional polytopes \(P_{0}\) and \(P_{1}\), then \(P\) is a strict Cayley polytope of order \(k\).
In Section \(5\) the author presents some algorithmic aspects, namely he gives three algorithms related to the local positivity for toric varieties. For a given polytope \(P\) and some local data the first algorithm allows to compute \(s(\mathcal{L}, x)\), the second one gives a lower and upper bound on the Seshadri constant in the case of toric surfaces and the last algorithm allows to check whether \(P\) is a Cayley polytope of order \(k\) with every edge of length at least \(k\). Illustrative examples are also delivered.

MSC:

14C20 Divisors, linear systems, invertible sheaves
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14N05 Projective techniques in algebraic geometry

Citations:

Zbl 1085.14516
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References:

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