a Weierstrass gap
/ VY-er-shtrahss /
Fix a single point p on a compact surface and ask: for which n is there a meromorphic function with a pole of order EXACTLY n at p and nowhere else? On the sphere every n >= 0 works (z^n has a pole of order n at infinity). But on a surface with holes, some orders are forbidden — there is no function with a pole of precisely that order at that one point. Those forbidden orders are the Weierstrass gaps, and they reveal hidden structure attached to each point.
Precisely, for a point p on a compact Riemann surface of genus g, consider the divisors n[p] and the dimensions l(n[p]) as n increases from 0. Each step n -> n+1 either keeps l the same (no NEW function with pole order exactly n+1 — a gap) or increases it by one (a new function appears — a non-gap or pole number). The Weierstrass Gap Theorem says there are EXACTLY g gaps, and they all occur among n = 1, 2, ..., 2g - 1. For a generic point the gaps are the first g positive integers 1, 2, ..., g (the 'expected' gap sequence). A point whose gap sequence deviates from this generic one is a Weierstrass point. Computationally: by Riemann-Roch, l(n[p]) - l((n-1)[p]) jumps iff l(K - n[p]) drops, i.e. iff there is a holomorphic differential vanishing to order exactly n - 1 at p — so the gaps are read off the vanishing orders of holomorphic 1-forms at p.
Why it matters: gaps encode how 'special' a point is and tie the local pole structure to the global genus (exactly g gaps, always). The non-gaps form a numerical semigroup, leading to the rich theory of Weierstrass points, which are finite in number and distinguished intrinsically — a canonical finite set attached to the surface, used to study automorphisms and moduli. Honesty caveat: 'Weierstrass point' has two related meanings — sometimes any point, described by its full gap sequence; sometimes specifically a point whose sequence is NON-generic. On the sphere (g = 0) there are no gaps at all and no Weierstrass points; the phenomenon is genuinely about positive genus. And the gap theorem is a direct consequence of Riemann-Roch applied point by point, not an independent miracle.
On a genus-2 curve there are exactly g = 2 gaps. At a generic point the gaps are {1, 2} and the non-gaps start 3, 4, 5, ... ; but at the six Weierstrass points (the branch points of the hyperelliptic cover) the gap sequence is {1, 3}: there IS a function with a double pole (the hyperelliptic x), which is impossible at a generic point — that exceptional 2 in the non-gaps is what marks the point as Weierstrass.
Generic genus-2 point: gaps {1,2}; a Weierstrass point: gaps {1,3} — a degree-2 function exists exactly there.
There are always exactly g gaps (a Riemann-Roch consequence, not magic), lying within 1..2g-1. Beware the dual usage of 'Weierstrass point': sometimes any point with its gap data, sometimes specifically a point with a NON-generic gap sequence. Genus 0 has no gaps at all.