On plane rational curves and the splitting of the tangent bundle

TL;DR

This paper investigates the splitting of the tangent bundle pullback for rational curves in projective planes, providing new bounds, classifications for generic points, and exploring cases with unbalanced splitting related to specific geometric configurations.

Contribution

It introduces new methods for determining splitting types, classifies these types for up to 7 points, and reveals infinitely many unbalanced cases for 9 generic points, connecting them to a semi-adjoint formula.

Findings

Complete classification for up to 7 points.

Existence of infinitely many unbalanced splittings for 9 points.

Connection between unbalanced splitting and semi-adjoint formula.

Abstract

Given an immersion $\phi: P^1 \to \P^2$, we give new approaches to determining the splitting of the pullback of the cotangent bundle. We also give new bounds on the splitting type for immersions which factor as $\phi: P^1 \cong D \subset X \to P^2$, where $X \to P^2$ is obtained by blowing up $r$ distinct points $p_i \in P^2$. As applications in the case that the points $p_i$ are generic, we give a complete determination of the splitting types for such immersions when $r \leq 7$. The case that $D^2=-1$ is of particular interest. For $r \leq8$ generic points, it is known that there are only finitely many inequivalent $\phi$ with $D^2=-1$, and all of them have balanced splitting. However, for $r=9$ generic points we show that there are infinitely many inequivalent $\phi$ with $D^2=-1$ having unbalanced splitting (only two such examples were known previously). We show that these new examples are related to a semi-adjoint formula which we conjecture accounts for all occurrences of unbalanced splitting when $D^2=-1$ in the case of $r=9$ generic points $p_i$. In the last section we apply such results to the study of the resolution of fat point schemes.