A lower bound for $$K^2_S$$ K S 2

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 \(K^2_S\ge -d(d-6)\). The bound is sharp, and \(K^2_S=-d(d-6)\) if and only if d is even, the linear system \(|H^0(S,{\mathcal {L}})|\) embeds S in a smooth rational normal scroll \(T\subset {\mathbb {P}}^5\) of dimension 3, and here, as a divisor, S is linearly equivalent to \(\frac{d}{2}Q\), where Q is a quadric on T.