Leading coefficients and effective range of asymptotic Betti-number polynomials

Determine, for a smooth projective complex curve C and a very ample line bundle L of sufficiently large degree, the leading coefficients and an effective range for the polynomial dependence on deg L of the graded Betti numbers in the overlap range where both relevant Koszul cohomology groups are nonzero.

Background

For sufficiently large degree, the graded Betti numbers of a curve’s section ring in the overlap range are known to be polynomials in the degree of the very ample line bundle. The degrees of these polynomials reflect geometric properties of the curve.

Before the results developed in the paper, the leading coefficients and an effective threshold beyond which the polynomial behavior holds were unresolved in general. The paper computes these quantities under specific Brill–Noether and cohomological hypotheses, but the general problem remains explicitly identified as open.