Debojyoti Bhattacharya
TIFR, Mumbai
May 2, 2024
On Brill-Noether locus over certain surfaces of general type in P^3.: Let $X$ be a smooth, projective, irreducible variety of dimension $n$ over $\mathbb C$ and $H$ be an ample divisor on $X$. Let $M_H:=M_{X,H}(r,c_1,...c_{\text{min}\{r,n\}})$ be the moduli space of rank $r, H$-stable vector bundles $ E$ on $X$ with fixed Chern classes $c_i(E)=c_i$. Brill-Noether theory deals with the geometry of subvarieties of $M_H$ whose support corresponds to points in $M_H$ admitting at least $k+1$ sections (called Brill-Nother locus and denoted by $\mathcal B^k_{r,H}$). In $2010$, as a systematic generalization of Brill-Noether theory of vector bundles over curves R.M Miro Roig and L.Costa constructed such subvarieties $\mathcal B^k_{r,H}$ of $M_H$ for $n \geq 2$ under higher cohomology vanishing hypothesis and investigated their geometry for some specific instances. After a brief literature survey of key aspects of the Brill-Noether theory of line bundles on curves followed by vector bundles on curves, we will discuss certain aspects of the Brill-Noether locus of rank $2$ stable bundles over very general hypersurfaces of degree at least $5$ in $\mathbb P^3$. If time permits, at the end of the talk we will also mention their relation with Ulrich and weakly Ulrich bundles on such surfaces.
TIFR, Mumbai
May 2, 2024
On Brill-Noether locus over certain surfaces of general type in P^3.: Let $X$ be a smooth, projective, irreducible variety of dimension $n$ over $\mathbb C$ and $H$ be an ample divisor on $X$. Let $M_H:=M_{X,H}(r,c_1,...c_{\text{min}\{r,n\}})$ be the moduli space of rank $r, H$-stable vector bundles $ E$ on $X$ with fixed Chern classes $c_i(E)=c_i$. Brill-Noether theory deals with the geometry of subvarieties of $M_H$ whose support corresponds to points in $M_H$ admitting at least $k+1$ sections (called Brill-Nother locus and denoted by $\mathcal B^k_{r,H}$). In $2010$, as a systematic generalization of Brill-Noether theory of vector bundles over curves R.M Miro Roig and L.Costa constructed such subvarieties $\mathcal B^k_{r,H}$ of $M_H$ for $n \geq 2$ under higher cohomology vanishing hypothesis and investigated their geometry for some specific instances. After a brief literature survey of key aspects of the Brill-Noether theory of line bundles on curves followed by vector bundles on curves, we will discuss certain aspects of the Brill-Noether locus of rank $2$ stable bundles over very general hypersurfaces of degree at least $5$ in $\mathbb P^3$. If time permits, at the end of the talk we will also mention their relation with Ulrich and weakly Ulrich bundles on such surfaces.