divisorn

Divisorn is a mathematical construct used in algebraic geometry to illustrate how divisibility phenomena interact with divisor theory on algebraic curves. In this framework, a divisorn on a smooth projective curve C over a field k is defined as a finite formal sum D = sum_P n_P [P], where P runs over the closed points of C and each coefficient n_P is an integer that is a multiple of a fixed positive integer d. The degree of a divisorn is deg(D) = sum_P n_P, which is always a multiple of d.

Divisorn arithmetic mirrors ordinary divisor arithmetic but with the fixed divisibility constraint. Addition is performed by adding coefficients componentwise, and the set of divisorns with coefficients in dZ forms a submonoid under this addition. Principal divisorns arise from nonzero rational functions on C in the usual way, and one can form a divisorn class group Div_d(C) = Div_d(C) / Prin(C), paralleling the classical divisor class group but restricted to divisorns. This construction provides a conceptual tool for examining how divisibility by d interacts with linear systems, morphisms, and pullbacks.

Useful properties include that on the projective line, any divisorn is a multiple of d times a divisor supported at points, and the degree remains a multiple of d. While divisorns are primarily a pedagogical device in expository contexts, they help clarify the relationships between divisors, linear equivalence, and the effect of fixed divisibility constraints on divisor-theoretic structures.

References to divisorn appear in introductory discussions of divisor theory and in pedagogical treatments that aim to isolate the role of divisibility in the arithmetic of divisors on curves.