On translated rank-\(2\) Brill-Noether loci on regular surfaces
type: Item
instance of: scholarly article
title: On translated rank-\(2\) Brill-Noether loci on regular surfaces
zbMATH Open document ID: 1487.14091
author: Laura Filimon
published in: Archiv der Mathematik
publication date: 2022-03-14T00:00:00Z
review text: Let \(X\) be a smooth complex projective variety of dimension \(n\) and \(H\) an ample divisor on \(X\), and let \(M_H:=M_H(r;c_1,\ldots,c_s)\) be the moduli space of vector bundles of rank \(r\) and Chern classes \(c_1,\ldots,c_s\) (\(s=\min\{r,n\}\)) on \(X\), which are slope-stable with respect to \(H\). One can define Brill-Noether loci in the same way as for curves by writing \(W_H^k:=W_H^k(r;c_1,\ldots,c_s)\) for the set of bundles \(E\in M_H\) with \(h^0(E)\ge k\). \textit{L. Costa} and \textit{R. M. Miró-Roig} [Forum Math. 22, No. 3, 411--432 (2010; Zbl 1190.14040)] showed that \(W_H^k\) can be given a structure of determinantal variety provided that \(h^i(E)=0\) for \(i\ge2\) and every \(E\in M_H\). Moreover, each non-empty irreducible component of \(W_H^k\) has dimension at least \(\rho^k_H:=\dim(M_H)-k(k-\chi(r;c_1,\cdots,c_s))\) and \(W_H^{k+1}\) is contained in the singular set of \(W_H^k\) whenever \(W_H^k\ne M_H\). The author of the current paper extends this study by considering the infinitesimal structure of \(W_H\) under the same hypothesis that \(h^i(E)=0\) for \(i\ge2\) and any \(E\in M_H\). She obtains a result similar to that for curves, namely that, if \(E\in W^k_h\setminus W^{k+1}_H\), the Zariski tangent space of \(W^k_H\) at \(E\) is isomorphic to \((Im(\mu_E))^\perp\), where \(\mu_E\) is the cup-product map \(H^0(E)\otimes H^{n-1}(E^*\otimes K_X)\to H^{n-1}(K_X\otimes E^*\otimes E)\). Moreover, if \(M_H\) is smooth at \(E\), then \(W^k_H\) is smooth and of the expected dimension \(\rho^k_H\) at \(E\) if and only if \(\mu_E\) is injective. Now let \(X\) be a smooth regular surface and let \({\mathcal C}_X\) denote the ample cone in \(\operatorname{Num}(X)\otimes {\mathbb R}\). Let \(\zeta\in\operatorname{Num}(X)\otimes {\mathbb R}\) be such that \(\zeta\equiv 2F-c_1\) for some divisor \(F\) on \(X\) and \(-(4c_2-c_1^2)\le\zeta^2<0\). A wall \(W^\zeta\) of type \((c_1,c_2)\) is defined as the set of all \(x\in{\mathcal C}_X\) such that \(x\c…
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14J26
Mathematics Subject Classification ID: 14J28
zbMATH DE Number: 7489070
zbMATH Keywords: vector bundles
zbMATH Keywords: Brill-Noether loci
zbMATH Keywords: projective surface
reviewed by: P. E. Newstead
MaRDI profile type: Publication
full work available at URL: s00013-021-01700-2
OpenAlex ID: W4213094190
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DOI: 10.1007/S00013-021-01700-2
description: scientific article; zbMATH DE number 7489070