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Generalized Euler sequence and toric varieties. (English) Zbl 0837.14042
Ciliberto, Ciro (ed.) et al., Classification of algebraic varieties. Algebraic geometry conference on classification of algebraic varieties, May 22-30, 1992, University of L’Aquila, L’Aquila, Italy. Providence, RI: American Mathematical Society. Contemp. Math. 162, 227-247 (1994).
A line bundle generated by global sections determines a mapping into projective space. In this paper we shall give a generalization of this correspondence: a mapping that leads to some torus embedding determined by some reducible divisor. Our starting point is the observation that many properties of the sheaf \({\mathcal O}_{\mathbb{P}^n} (1)\) follow immediately from the fact that the direct sum of \(n + 1\) copies of this sheaf is the nontrivial extension of the tangent sheaf by the structural one. Dualizing the corresponding short exact sequence we obtain a sequence of the form: \[ 0 \to \Omega_{\mathbb{P}^n} \to \oplus {\mathcal O}_{\mathbb{P}^n} (-1) \to {\mathcal O}_{\mathbb{P}^n} \to 0, \] where \(\Omega_{\mathbb{P}^n}\) denotes the cotangent sheaf, so we may identify the fibre of the trivial bundle with the space \(H^1 (\Omega_{\mathbb{P}^n})\). For any nonsingular variety one may find a short exact sequence of the similar form that has functorial properties (see definition 2.1). It turns out that torus embeddings are the only varieties for which the sheaves obtained as such extensions are the direct sums of the invertible sheaves which correspond to nonsingular divisors intersecting transversally (theorem 3.1). In this context it is natural to ask which divisors, on an arbitrary nonsingular variety \(Y\), are pullbacks of the distinguished divisors on torus embeddings. For a vast divisor (definition 4.3) there exists a mapping leading to a torus embeddding and such that this divisor is the pullback of the distinguished one on the torus embedding.
For the entire collection see [Zbl 0791.00020].

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14C20 Divisors, linear systems, invertible sheaves