Focal loci of families and the genus of curves on surfaces

Chiantini, Luca; Lopez, Angelo

doi:10.1090/S0002-9939-99-05407-6

Focal loci of families and the genus of curves on surfaces
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by Luca Chiantini and Angelo Felice Lopez PDF

Proc. Amer. Math. Soc. 127 (1999), 3451-3459 Request permission

Corrigendum: Proc. Amer. Math. Soc. 137 (2009), 3951-3951.

Abstract:

In this article we apply the classical method of focal loci of families to give a lower bound for the genus of curves lying on general surfaces. First we translate and reprove Xu’s result that any curve $C$ on a general surface in $\mathbb {P}^{3}$ of degree $d \geq 5$ has geometric genus $g > 1 + \operatorname {deg} C (d - 5) / 2$. Then we prove a similar lower bound for the curves lying on a general surface in a given component of the Noether-Lefschetz locus in $\mathbb {P}^{3}$ and on a general projectively Cohen-Macaulay surface in $\mathbb {P}^{4}$.

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