Martens-Mumford's Theorems for Brill-Noether Schemes arising from Very Ample Line Bundles

TL;DR

This paper extends Martens-Mumford's theorems to Brill-Noether schemes derived from very ample line bundles, providing new insights into their tangent spaces and dimension bounds on algebraic curves.

Contribution

It proves Martens' theorem for specific Brill-Noether schemes associated with very ample line bundles, identifying conditions for maximum dimension attainment.

Findings

Describes tangent spaces of V^r_d(L) schemes.

Establishes dimension bounds for these schemes.

Identifies curves where schemes reach maximum dimension.

Abstract

Tangent Spaces of V^r_d(L), Specific subschemes of C_d arising from various line bundles on C, are described. Then we proceed to prove Martense Theorem for these schemes, by which we determine curves C, which for some very ample line bundle L on C and some integers r and d with d\leq h^{0}(L)-2, the subscheme V^r_d(L) might attain its maximum dimension.