Mathematical Research Letters
Volume 23 (2016)
Pointed Castelnuovo numbers
Pages: 389 – 404
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.