## View abstract

### 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. document.getElementById('cloak4c7122173285a5497b7c8393aee43844').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy4c7122173285a5497b7c8393aee43844 = 'l&#101;&#111;n&#97;rd&#111;.r&#111;&#97;' + '&#64;'; addy4c7122173285a5497b7c8393aee43844 = addy4c7122173285a5497b7c8393aee43844 + 'c&#105;m&#97;t' + '&#46;' + 'mx'; var addy_text4c7122173285a5497b7c8393aee43844 = 'l&#101;&#111;n&#97;rd&#111;.r&#111;&#97;' + '&#64;' + 'c&#105;m&#97;t' + '&#46;' + 'mx';document.getElementById('cloak4c7122173285a5497b7c8393aee43844').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy4c7122173285a5497b7c8393aee43844 + '\'>'+addy_text4c7122173285a5497b7c8393aee43844+'<\/a>';

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 https://arxiv.org/pdf/2003.02727.pdf) .

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