On plane rational curves and the splitting of the tangent bundle

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.