Projective Q-factorial toric varieties covered by lines

2005/12/16 by C. Casagrande, Cinzia Casagrande, Casagrande, C. +3
Computer Science · Mathematics · #14M25 #52B20 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Polynomial and algebraic computation #math.AG #math.CO #msc:14M25 #msc:52B20

paper · pdf · doi:10.48550/arxiv.math/0512385

24 pages. Refereed version, to appear in CCM (Communications in Contemporary Mathematics)

openalex publication_date 2005/12/16 · arxiv created 2007/01/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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).

Citations

• On covering and quasi-unsplit families of rational curves

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