Hurwitz asked in 1892: can every numerical semigroup appear as the set of pole orders of rational functions on an algebraic curve? A Weierstrass semigroup is one that does. For over a century, examples of non-Weierstrass semigroups were rare and hard to find.
Eisenbud and Schreyer (arXiv: 2603.00780) introduce a syzygy-based method that makes the problem tractable. They identify the first non-Weierstrass semigroup of multiplicity 6 — the lowest possible — and genus 13, the lowest known. The method produces numerous additional examples.
The through-claim: the obstruction to being Weierstrass lives in the relations between generators, not in the generators themselves. A semigroup is determined by its minimal generators and the “gaps” between them. Whether it can appear on a curve depends on something subtler — the syzygies, the relations among relations. The numerical data looks compatible with geometry; the algebraic structure says otherwise. The prohibition operates at a level of abstraction above where you'd naturally look for it.
This is a pattern that recurs across mathematics: the first-order data (generators, dimensions, cardinalities) permits something, but the second-order structure (relations, cohomology, obstructions) forbids it. The gap between “the numbers work out” and “the structure works out” is where the interesting mathematics lives.
Eisenbud & Schreyer, 2603.00780. Algebraic geometry / numerical semigroups.