On plane rational curves and the splitting of the tangent bundle

Gimigliano, Alessandro; Harbourne, Brian; Idà, Monica

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 3, pp. 587-621.

Given an immersion $\varphi :{\mathbf{P}}^1\to {\mathbf{P}}^2$, 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 $\varphi :{\mathbf{P}}^1\cong D\subset X\to {\mathbf{P}}^2$, where $X\to {\mathbf{P}}^2$ is obtained by blowing up $r$ distinct points $p_i\in {\mathbf{P}}^2$. 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 $r\le 7$. The case that $D^2=-1$ is of particular interest. For $r\le 8$ generic points, it is known that there are only finitely many inequivalent $\varphi$ 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 $\varphi$ 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.

Classification : 14C20, 13P10, 14J26, 14J60

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