Mathematical Research Letters

Volume 23 (2016)

Number 2

Pointed Castelnuovo numbers

Pages: 389 – 404



Gavril Farkas (Institut für Mathematik, Humboldt-Universität zu Berlin, Germany)

Nicola Tarasca (Department of Mathematics, University of Utah, Salt Lake City, Ut., U.S.A.)


The classical Castelnuovo numbers count linear series of minimal degree and fixed dimension on a general curve, in the case when this number is finite. For pencils, that is, linear series of dimension one, the Castelnuovo numbers specialize to the better known Catalan numbers. Using the Fulton–Pragacz determinantal formula for flag bundles and combinatorial manipulations, we obtain a compact formula for the number of linear series on a general curve having prescribed ramification at an arbitrary point, in the case when the expected number of such linear series is finite. The formula is then used to solve some enumerative problems on moduli spaces of curves.


Brill–Noether theory, enumerative geometry on a general curve

2010 Mathematics Subject Classification

Primary 14Q05. Secondary 14H51.

Accepted 23 November 2015

Published 6 June 2016