Quartic 3-fold: Pfaffians, vector bundles, and half-canonical curves
The paper under review belongs to a series of articles by the authors and \textit{A.S. Tikhomirov} [J. Algebr. Geom. 10, 37--62 (2001; Zbl 0987.14028); Doc. Math., J. DMV 5, 23--47 (2000; Zbl 0938.14021); Int. Math. Res. Not. 51, 2747--2778 (2003; Zbl 1048.14028)], in which arithmetically Cohen-Macaulay (ACM) bundles on certain Fano threefolds are investigated, especially by studying an associated Abel-Jacobi map.NEWLINENEWLINENEWLINENEWLINEHere the main interest is in the moduli space \({\mathcal M}_X(2;-1,6)\) of rank two bundles \(E\) with \(c_1(E)=-1\), \(c_2(E)=6\)NEWLINEon a generic quartic threefold \(X\subset \mathbb P^4\). An open component \({\mathcal M}_X\) of this moduli space is formed by ACM bundles (i.e. \(H^i(X,E(j))=0\) for \(i=1,\,2\) and \(j\in \mathbb Z\)). By a general result of \textit{A. Beauville} [Mich. Math. J. 48, 39--64 (2000; Zbl 1076.14534)], such a bundle is given by a sequence \(0\to \mathcal O_{\mathbb P^4}(-1)^8\overset {M} {\to}\mathcal O_{\mathbb P^4}{^8} {E} 0\) where \(M\) is an \(8\times 8\) skew-symmetric matrix whose pfaffian is the equation of \(X\). NEWLINENEWLINENEWLINE The authors prove that there is a \(7\)-parameter family of representations of \(X\) as such a pfaffian (hence dim\({\mathcal M}_X\)=7). The ACM curves associated to such bundles by the Serre correspondence are characterized as linear sections rank 4 locus \(Z\subset \mathbb P(\bigwedge^2\mathbb{C}^7)\) in the space of \(7\times 7\) skew-symmetric matrices (by Beauville, they are half-canonical ACM curves in \(\mathbb P^4\) of degree 14). Their locus in the Hilbert scheme and the moduli space of genus 15 curves is studied. The Abel-Jacobi map for such curves factors through \(M_X\) via the Serre correspondence with fibers \(\mathbb P^7\) over a point of \({\mathcal M}_X\).NEWLINEThe resulting map \({\mathcal M}_X \to J^2(X)\) is shown to be quasi-finite and unramified.
scientific article; zbMATH DE number 1700848