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Journal of the London Mathematical Society Advance Access published online on July 8, 2008
Journal of the London Mathematical Society, doi:10.1112/jlms/jdn038
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© 2008 London Mathematical Society

Higher ramification and varieties of secant divisors on the generic curve

Gavril Farkas

Department of Mathematics
University of Texas
Austin, TX 78712
USA
gfarkas@math.utexas.edu

For a smooth projective curve, the cycles of e-secant k-planes are among the most studied objects in classical enumerative geometry, and there are well-known formulas due to Castelnuovo, Cayley and MacDonald concerning them. Despite various attempts, surprisingly little is known about the enumerative validity of such formulas. The aim of this paper is to clarify this problem in the case of the generic curve C of given genus. We determine precisely under which conditions the cycle of e-secant k-planes is non-empty, and we compute its dimension. We also precisely determine the dimension of the variety of linear series on C carrying e-secant k-planes.

The Research is partially supported by an Alfred P. Sloan Fellowship, the NSF Grants DMS-0450670 and DMS-0500747 and a 2006 Texas Summer Research Assignment.

2000 Mathematics Subject Classification 14H10, 14C20.

Received April 17, 2007; revised January 24, 2008;
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