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Session S35 - Moduli Spaces in Algebraic Geometry and Applications

Thursday, July 15, 15:40 ~ 16:00 UTC-3

On the Segre Invariant for Rank Two Vector Bundles on  $\mathbb{P}^2$

Leonardo Roa Leguizamón

Universidad Michoacana de San Nicolas de Hidalgo, Mexico   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We extend the concept of the Segre Invariant to vector bundles on a surface $X$. For a vector bundle $E$ of rank 2 on $X$, the Segre invariant is defined as the minimum of the differences between the slope of $E$ and the slope of all line subbundles of $E$.  This invariant defines a semicontinuous function on the families of vector bundles on $X$. Thus, the Segre invariant gives a stratification of the moduli space $M_{X,H} (2; c_1, c_2)$ of $H$−stable vector bundles of rank 2 and fixed Chern classes $c_1$ and $c_2$ on the surface $X$ into locally closed subvarieties $M_{X,H} (2; c_1, c_2; s)$ according to the value of $s$. For $X=\mathbb{P}^2$ we determine what numbers can appear as the Segre Invariant of a rank $2$ vector bundle with given Chern's classes. The irreducibility of strata with fixed Segre invariant is proved and its dimensions are computed. Finally, we present applications to the Brill-Noether Theory for rank $2$ vector bundles on $\mathbb{P}^2.$ (see .

Joint work with H. Torres-López and A.G. Zamora (Universidad Autónoma de Zacatecas, Mexico).

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