Clifford's theorem on special divisors

For a special divisor D on a smooth projective curve, Clifford's theorem states that h⁰(D) ≤ deg(D)/2 + 1. Equality is tightly constrained, with exceptions associated with canonical divisors and hyperelliptic curves.

Connect

Riemann–Roch theorem for curves: A theorem relating a divisor's sections to its degree, the curve's genus, and sections of its canonical complement. Its formula turns the specialness condition into the numerical bound.
Clifford index: An invariant measuring the smallest degree-minus-section excess among special divisors on a curve. It packages the gap between Clifford's bound and actual sections.
William Kingdon Clifford: A British mathematician and philosopher whose work included a theorem on special divisors of algebraic curves. The theorem bears his name and comes from his work on curve divisors.
Clifford's theorem for vector bundles: A generalization of Clifford-type section bounds from line bundles to vector bundles on curves. It extends the same style of inequality beyond the divisors covered here.
Serre duality: A duality identifying cohomology of a line bundle with the dual of cohomology of its canonical inverse. It identifies the residual divisor whose sections pair with those of D.
Hyperelliptic linear series: Linear series on a hyperelliptic curve, often obtained from multiples of its degree-two pencil. Multiples of the hyperelliptic pencil attain equality in Clifford's inequality.
Brill–Noether theory: The study of linear series on algebraic curves and the dimensions of their parameter spaces. Clifford's bound is a foundational constraint on its linear series.
Brill–Noether theorem: A theorem predicting dimensions of spaces of linear series on a general curve. It gives a dimension prediction rather than Clifford's bound for each special divisor.
Special divisor: A divisor D on a smooth projective curve for which the first cohomology of its line bundle is nonzero. This is the hypothesis that makes Clifford's inequality apply.
Trigonal curve: A smooth curve admitting a degree-three map to the projective line. Its degree-three pencil illustrates a low-degree special series near the bound.