On translated rank-\(2\) Brill-Noether loci on regular surfaces
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 \(x\cdot\zeta=0\). A polarisation \(H\) is said to be above (below) the wall if \(\zeta.H>0\) (\(\zeta.H<0\)). Define \(E_\zeta(c_1,c_2)\) to be the set of all locally free sheaves \(V\) of rank \(2\) given by non-trivial extensions of \({\mathcal O}_X(c_1-F)\otimes{\mathcal I}_Z\) by \({\mathcal O}_X(F)\) for some divisor \(F\) such that \(2F-c_1\equiv\zeta\) and \(Z\) is a locally complete intersection \(0\)-cycle of length \(c_2+\frac{\zeta^2-c_1^2}4\). Now let \(H^2(X,{\mathcal O}_X)=0\) and suppose that \(4c_2-c_1^2>0\) and that \(H\) lies in a chamber adjacent to the non-empty wall \(W^\zeta\) and below the wall. Suppose further that \(H^2(X,{\mathcal O}(c_1-2L))=0\) for all \(L\equiv\frac12(\zeta+c_1)\). Then (Theorem 3.3) \(E_\zeta(c_1,c_2)\) is the disjoint union of the translated Brill-Noether loci \(W^1_H(\tilde{c}_1,\tilde{c}_2)\otimes L\) with \(L\equiv\frac12(\zeta+c_1)\), where \(\tilde{c}_i(E)=c_1(E\otimes L^{-1})\). In particular, in the case of a Hirzebruch surface, \(E_\zeta(c_1,c_2)=W^1_H(\tilde{c}_1,\tilde{c}_2)\otimes L\) for \(L=\frac12(\zeta+c_1)\) and this translated Brill-Noether locus is non-empty. Finally, under the hypotheses of Theorem 3.3, \(W^1_H(\tilde{c}_1,\tilde{c}_2)\) is smooth of the expected dimension.
scientific article; zbMATH DE number 7489070
scientific article