On plane rational curves and the splitting of the tangent bundle
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 12 (2013) no. 3, p. 587-621

Given an immersion ϕ:P 1 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 ϕ:P 1 DXP 2 , where XP 2 is obtained by blowing up r distinct points p i 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 r7. The case that D 2 =-1 is of particular interest. For r8 generic points, it is known that there are only finitely many inequivalent ϕ 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 ϕ 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.

Published online : 2019-02-21
Classification:  14C20,  13P10,  14J26,  14J60
@article{ASNSP_2013_5_12_3_587_0,
     author = {Gimigliano, Alessandro and Harbourne, Brian and Id\`a, Monica},
     title = {On plane rational curves and the splitting of the tangent bundle},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 12},
     number = {3},
     year = {2013},
     pages = {587-621},
     zbl = {06232457},
     mrnumber = {3137457},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2013_5_12_3_587_0}
}
Gimigliano, Alessandro; Harbourne, Brian; Idà, Monica. On plane rational curves and the splitting of the tangent bundle. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 12 (2013) no. 3, pp. 587-621. http://www.numdam.org/item/ASNSP_2013_5_12_3_587_0/

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