Divisors on Riemann Surfaces

A rather simple concept of a divisor for a Riemann surface is supported. Here the divisors are realised as finite sums of points with integer coefficients. The type name for these Riemann surface divisors is DivRieSrfElt. The main application of these divisors is to the computation of the Abel–Jacobi map.

Divisor(S, V): SeqEnum[RieSrfPt], SeqEnum[RngIntElt] -> DivRieSrfElt

Given a sequence \(S\) of points \(P_i\) belonging to Riemann surface \(X\) and a sequence \(V\) of integers \(n_i\), construct the formal divisor \(\sum_i n_iP_i\).

ZeroDivisor(X): RieSrfElt -> DivRieSrfElt

Construct the zero divisor for the Riemann surface \(X\).

RiemannSurface(D): DivRieSrfElt -> RieSrf

The Riemann surface associated with the divisor \(D\) is returned.

Support(D): DivRieSrfElt -> SeqEnum[RieSrfPt], SeqEnum[RngIntElt]

Given a divisor \(D= \sum_i n_iP_i\) with all \(n_i\) nonzero, the sequence of points \(P_i\) and the sequence of their multiplicities \(n_i\) are returned.

Degree(D): DivRieSrfElt -> RngIntElt

The sum of the multiplicities \(n_i\) of the points \(P_i\) supporting the divisor \(D = \sum_I N_Ip_I\) is returned.

RandomDivisor(X, d): RieSrf, RngIntElt -> RieSrfDivElt

Ht  : RngIntElt                    Default: 10^5
Zero: BoolElt                      Default: true

Given a Riemann surface \(X\) and a positive integer \(d\) return a random divisor for \(X\) of degree \(d\). The maximum size of the coefficients can be bounded by assigning a positive integer to the parameter Ht. If parameter Zero is set to true, the degree of the returned divisor will be zero.