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Projective Q-factorial toric varieties covered by lines

C. Casagrande·S. Di Rocco
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Abstract

The main result of this paper is a structural theorem for projective Q-factorial toric varieties X in P^N, covered by lines. We prove that there exists a toric fibration f: X -> Z, locally trivial in the Zariski topology, with fiber a product of projective joins. All lines in X intersecting the open subset isomorphic to the torus, are contained in some fiber of f. This characterization has a geometrical application to dual defective toric varieties, and a combinatorial application to discriminants of lattice subsets. We prove that X has positive dual defect if and only if it has an elementary extremal contraction of fiber type, whose general fiber is a projective join with dual defect bigger than its codimension in X. Turning to combinatorics, we characterize lattice subsets A with discriminant D_A equal to one, under suitable assumptions on the polytope Conv(A).

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C. Casagrande, S. Di Rocco. 2007-01-19. Projective Q-factorial toric varieties covered by lines. https://arxiv.org/abs/math/0512385

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