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 P3 of degree d 5 has geometric genus g > 1 + degC(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 P3 and on a general projectively Cohen-Macaulay surface in P4.
Chiantini, L., Lopez, A.F. (1999). Focal loci of families and the genus of curves on surfaces. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 127(11), 3451-3459 [10.1090/S0002-9939-99-05407-6].
Focal loci of families and the genus of curves on surfaces
CHIANTINI, LUCA;
1999-01-01
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 P3 of degree d 5 has geometric genus g > 1 + degC(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 P3 and on a general projectively Cohen-Macaulay surface in P4.
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https://hdl.handle.net/11365/36388
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