Focal loci of families and the genus of curves on surfaces

Details Focal loci of families and the genus of curves on surfaces Focal loci of families and the genus of curves on surfaces

1999-03-08

arxiv.org/abs/math/9903044v3

PAMS n. 127/12

Mathematics

Algebraic Geometry

Plain TeX, 11 pages, to appear on PAMS. Minor corrections in the statement of 1.3 and in remark 3.4

Scientific paper

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 P^3 of degree d>4 has geometric genus g > 1 + 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 P^3 and on a general projectively Cohen-Macaulay surface in P^4.

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