  For a generic compact Riemann surface the theta function is at every point on
the Jacobian equal to its first Taylor term, up to a holomorphic change of
local coordinates and multiplication by a local holomorphic unit. More
generally, any Brill-Noether locus of twisted stable vector bundles on a smooth
projective curve is at every point L locally \'etale isomorphic with its
tangent cone if the Petri map at L is injective. This assumption has various
consequences for Brill-Noether loci: positive answers to the monodromy
conjecture for generalized theta divisors and to questions of Schnell-Yang on
log resolutions and Whitney stratifications, and formulas for local
b-functions, log canonical thresholds, topological zeta functions, and minimal
discrepancies.
