Combinatorial expression of the fundamental second kind differential on an algebraic curve

Eynard, Bertrand

Annales de l’Institut Henri Poincaré D, Tome 9 (2022) no. 2, pp. 219-238.

The zero locus of a bivariate polynomial $P(x,y)=0$ defines a compact Riemann surface $\Sigma$. The fundamental second kind differential is a symmetric $1\otimes 1$-form on $\Sigma \times \Sigma$ that has a double pole at coinciding points and no other pole. As its name indicates, this is one of the most important geometric objects on a Riemann surface. Here we give a rational expression in terms of combinatorics of the Newton's polygon of $P$, involving only integer combinations of products of coefficients of $P$. Since the expression uses only combinatorics, the coefficients are in the same field as the coefficients of $P$.

Classification : 14-XX
Keywords: algebraic geometry, Riemann surfaces, Newton polygon