Characteristic numbers of ellitpic space curves with fixed j-invariant

TL;DR

This paper develops recursive formulas to count elliptic space curves with fixed j-invariant under tangency conditions, providing explicit formulas for low-dimensional cases and supporting numerical examples.

Contribution

It introduces effective recursion formulas for counting elliptic curves with fixed j-invariant in projective space, including explicit solutions for dimensions up to five.

Findings

Derived effective recursion formulas for elliptic curves counting

Provided explicit formulas for dimensions ≤ 5

Included numerical examples and a C++ implementation

Abstract

We solve the problem of counting elliptic curves with fixed j-invariant in projective space with tangency conditions. This is equivalent to couting rational nodal curves with condition on the node of the image. The solution is given in the form of effective recursions. We give explicit formulas when the dimension of the ambient projective space is at most 5. Many numerical examples are provided. A C++ program implementing most of the recursions is available on the author's webpage (see \cite{dn}).