New component of the moduli space \(M\)(2;0,3) of stable vector bundles on the double space \(\mathbb{P}^3\) of index two (Q1868209)

Let \(X\) be a smooth quartic double solid, i.e., \(\pi: X \rightarrow \mathbb{P}^3\) is a double covering branched along a smooth quartic \(Q\subset\mathbb{P}^3\). Hence, \(X\) is a Fano variety of index 2. Recently, moduli of rank two vector bundles with small Chern classes on these varieties have attracted some interest; \textit{M. Szurek} and \textit{J. A. Wisniewski} [Rev. Roum. Math. Pures Appl. 38, No.7--8, 729--741 (1993; Zbl 0816.14015)] constructed such moduli for \(c_1=-1\), \(c_2=2\) and the reviewer for \(c_1=0\), \(c_2=2\) [in: Proc. Berlin Math. Soc. 1997--2000, 181--200 (2001; Zbl 1082.14045)]. In the paper under review, the author applies the techniques developed in several joint papers with \textit{D. Markushevich} [J. Algebr. Geom. 10, 37--62 (2001; Zbl 0987.14028); Doc. Math., J. DMV 5, 23--47 (2000; Zbl 0938.14021)] to study moduli of bundles on certain Fano varieties in terms of an associated Abel-Jacobi map. More precisely, the main interest here is the moduli space \(M(2;0,3)\) of stable vector bundles of rank two with Chern classes \(c_1=0\), \(c_2=3\), which are related via Serre correspondence to an open set \(H\) of the family of elliptic quintics on \(X\). The author studies the associated Abel-Jabobi map \(\Phi_H: H\rightarrow J(X)\). He shows that \(\Phi_H\) factors through \(M(2;0,3)\) and that this gives the Stein factorization of the Abel-Jacobi map. Furthermore, he proves that the image of \(\Phi_H\) is a translate of the theta divisor on \(J(X)\), using a criterion of \textit{G. Welters} [``Abel-Jacobi isogenies for certain types of Fano threefolds'', Math. Centre Tracts, 141, Mathematisch Centrum, Amsterdam (1981; Zbl 0474.14028)]. It should be noted that an extended version has recently been published as a joint paper with \textit{D. Markushevich} [Int. Math. Res. Not. 51, 2747--2778 (2003; Zbl 1048.14028)]; containing detailed proofs and further results, showing irreducibility of \(H\) and computing the degree of the Stein factorization as \(84\).