Linear Series on General Curves With Prescribed Incidence Conditions

Abstract

Using degeneration and Schubert calculus, we consider the problem of computing the number of linear series of given degree d and dimension r on a general curve of genus g satisfying prescribed incidence conditions at n points. We determine these numbers completely for linear series of arbitrary dimension when d is sufficiently large, and for all d when either $r=1$ or $n=r+2$ . Our formulas generalise and give new proofs of recent results of Tevelev and of Cela, Pandharipande and Schmitt.

Tevelev degrees and Hurwitz moduli spaces

“…Cavalieri, Markwig and Ranganathan [7] have proposed a tropical calculation of the Tevelev degrees . Another calculation of the Tevelev degrees is given by Farkas and Lian in [13] via degenerations and the Schubert calculus. Farkas and Lian analyse a different fiber of than we do, but in the end recover the same formulas.…”

“…Together, the papers [5] and [13] place the study of Tevelev degrees in a much wider setting. The common theme is that the Tevelev degrees are surprisingly computable in closed form.…”

A Generalized RSK for Enumerating Linear Series on n-pointed Curves

“…In a recent paper [9], Farkas and Lian provide enumerative formulas for the number of maps from a curve C to a complex projective space P r with specified incidence conditions. In particular, let C be a general curve of genus g, and let x 1 , .…”

“…For sufficiently large d and setting n = (dr + d + r − rg)/r, it was shown in [9] that (1) L g,r,d = (r + 1) g .…”

“…Example 1.2. The following is an L-tableau with parameters (4,3,9). We write the "red" numbers as black font with a red shaded background for clarity.…”