A lower bound for K²_S

2016/01/25 by Vincenzo Di Gennaro, Di Gennaro, Vincenzo, Davide Franco +1
Mathematics · #14J99 #14M20 #14N15 #51N35 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1601.06698

openalex publication_date 2016/01/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (S,\mathcal L) be a smooth, irreducible, projective, complex surface, polarized by a very ample line bundle \mathcal L of degree d > 35. In this paper we prove that K²_S≥ -d(d-6). The bound is sharp, and K²_S=-d(d-6) if and only if d is even, the linear system |H⁰(S,\mathcal L)| embeds S in a smooth rational normal scroll T⊂ \mathbb P⁵ of dimension 3, and here, as a divisor, S is linearly equivalent to (d)/(2)Q, where Q is a quadric on T.

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