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Focal loci of families and the genus of curves on surfaces

Authors:	Luca Chiantini Angelo Felice Lopez

Institution:	Dipartimento di Matematica, Università di Siena, Via del Capitano 15, 53100 Siena, Italy ; Dipartimento di Matematica, Università di Roma Tre, Largo San Leonardo Murialdo 1, 00146 Roma, Italy

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 on a general surface in $\mathbb{P}^{3}$ of degree $d \geq 5$ has geometric genus $g > 1 + \hbox {deg} C (d - 5) / 2$ . 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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