Naive Conjecture on interpolation for Brill–Noether curves

Determine whether Brill–Noether curves of degree d and genus g in projective space Pr interpolate ((r+1)d − (r−3)(g−1))/(r−1) general points, i.e., the number predicted by the Brill–Noether dimension count.

Background

The Brill–Noether Theorem identifies a unique component M(g, r, d)BN of the moduli space of curves in Pr consisting of general curves when ρ(g, r, d) ≥ 0, with dimension (r+1)d − (r−3)(g−1). A heuristic dimension-drop argument suggests that imposing each general point condition reduces dimension by r−1, leading to an expected number of general points that such curves should interpolate.

This expectation is formulated as a conjecture predicting that Brill–Noether curves interpolate exactly the number of points given by dividing the Brill–Noether dimension by r−1. Subsequent sections discuss counterexamples and a complete resolution identifying the precise exceptional cases, but the conjecture is explicitly stated in the text as such.