A lower bound for χ(\mathcal O_S)

2021/02/16 by Di Gennaro, Vincenzo
#14N15 #51N35 #Algebraic Geometry (math.AG) #FOS: Mathematics #Primary 14J99 #Secondary 14M20

paper · doi:10.48550/arxiv.2102.08285

Abstract

Let (S,\mathcal L) be a smooth, irreducible, projective, complex surface, polarized by a very ample line bundle \mathcal L of degree d > 25. In this paper we prove that χ(\mathcal O_S)≥ -(1)/(8)d(d-6). The bound is sharp, and χ(\mathcal O_S)=-(1)/(8)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. Moreover, this is equivalent to the fact that the general hyperplane section H∈ |H⁰(S,\mathcal L)| of S is the projection of a curve C contained in the Veronese surface V⊆ \mathbb P⁵, from a point x∈ V\backslash C.

Related

• A lower bound for K²_S
• Rational vs transcendental points on analytic Riemann surfaces
• On smooth surfaces in P4 containing a plane curve
• On the k-normality of some projective manifolds
• On The Projective Normality of Smooth Surfaces of degree nine