2011/02/05 by Alessandro Gimigliano, Gimigliano, Alessandro, Brian Harbourne +3 · 1 citation
Computer Science · Mathematics · #13P10 #14C20 #14J26 #14J60 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Polynomial and algebraic computation #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.1102.1093
openalex publication_date 2011/02/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given an immersion ϕ: P1 → ¶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: P1 ≅ D ⊂ X → P2, where X → P2 is obtained by blowing up r distinct points pi ∈ P2. As applications in the case that the points pi are generic, we give a complete determination of the splitting types for such immersions when r ≤ 7. The case that D2=-1 is of particular interest. For r ≤8 generic points, it is known that there are only finitely many inequivalent ϕ with D2=-1, and all of them have balanced splitting. However, for r=9 generic points we show that there are infinitely many inequivalent ϕ with D2=-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 D2=-1 in the case of r=9 generic points pi. In the last section we apply such results to the study of the resolution of fat point schemes.