A lower bound for $$\chi ({\mathcal {O}}_S)$$

Abstract Let $$(S,{\mathcal {L}})$$ ( S , L ) be a smooth, irreducible, projective, complex surface, polarized by very ample line bundle $${\mathcal {L}}$$ of degree $$d > 25$$ d > 25 . In this paper we prove that $$\chi (\mathcal O_S)\ge -\frac{1}{8}d(d-6)$$ χ O ≥ - 1 8 6 The bound is sharp, and ({\mathcal {O}}_S)=-\frac{1}{8}d(d-6)$$ = if only d even, the linear system $$|H^0(S,{\mathcal {L}})|$$ | H 0 embeds S in smooth rational normal scroll $$T\subset {\mathbb {P}}^5$$ T ⊂ P 5 dimension 3, here, as divisor, linearly equivalent to $$\frac{d}{2}Q$$ 2 Q , where Q quadric on T Moreover, fact general hyperplane section $$H\in |H^0(S,{\mathcal ∈ projection curve C contained Veronese surface $$V\subseteq V ⊆ from point $$x\in V\backslash C$$ x \ C