On the projective genus of surfaces

Sabatino, Pietro

Bollettino della Unione matematica italiana, Série 8, 9B (2006) no. 2, pp. 311-317

Résumé

Let $X \subset \mathbb{P}^N$ be a smooth irreducible non degenerate surface over the complex numbers, $N \geq 4$. We define the projective genus of $X$, denoted by $PG(X)$, as the geometric genus of the singular curve of the projection of $X$ from a general linear subspace of codimension four. Denote by $g(X)$ the sectional genus of $X$. In this paper we conjecture that the only surfaces for which $PG(X) = g(X) - 1$ are the del Pezzo surface in $\mathbb{P}^4$, in $\mathbb{P}^5$ and a conic bundle of degree 5 in $\mathbb{P}^4$. We prove that for $N \geq 5$ if $PG(X) = g(X) - 1 + \lambda$, $\lambda$ a non negative integer, then $g(X) \leq \lambda + 1 + \alpha$ where $\alpha = -2$ for a scroll and $\alpha = 0$ otherwise, and deduce the conjecture for $N \geq 5$ from this statement.