Counting projections of rational curves

TL;DR

This paper investigates the finite number of rational curves of a given degree that project onto two general rational curves in different projective spaces, using intersection theory and Bott residue formula for computation.

Contribution

It establishes conditions under which the number of such projecting rational curves is finite and provides a method to compute this number explicitly.

Findings

Number of projecting rational curves is finite under certain conditions.

Explicit computation of the number using intersection theory and Bott residue formula.

Provides a new approach to counting projections of rational curves.

Abstract

Given two general rational curves of the same degree in two projective spaces, one can ask whether there exists a third rational curve of the same degree that projects to both of them. We show that, under suitable assumptions on the degree of the curves and the dimensions of the two given ambient projective spaces, the number of curves and projections fulfilling the requirements is finite. Using standard techniques in intersection theory and the Bott residue formula, we compute this number.