On maximally non-factorial nodal Fano threefolds (Q6620356)

In this paper, the authors study maximally non-factorial nodal Fano threefolds. Recall that a nodal Fano threefold \(X\) is Fano threefolds with at worst isolated ordinary double points (nodes). The Picard group and class group of \(X\) are torsion-free of finite rank. We say \(X\) is maximally non-factorial if\N\begin{align*}\N\mathrm{ rk}\ Cl(X) -rk\ \mathrm{Pic}(X) = |\mathrm{Sing}(X)|. \N\end{align*}\N\NInspired by the recent advances of maximally non-factorial nodal Fano threefold and derived category of coherent sheaves for singular varieties, the authors propose the following question:\N\begin{align*}\N\text{ Classify all maximally non-factorial nodal Fano threefolds. }\N\end{align*}\N\NThe authors give a partial answer to this question, they classify maximally non-factorial nodal Fano threefolds of Picard rank one that have exactly one singular point (node),\N\begin{itemize}\N\item the nodal Fano threefold \(X\) has one node,\N\item the rank of the Picard group Pic(\(X\)) is one,\N\item the rank of the class group Cl(\(X\)) is two.\N\end{itemize}\N\NThe main result of the paper is Theorem 1.3, the authors show that\N\begin{align*}\N\text{ There are exactly 17 types of non-factorial Fano threefolds of Picard rank one with one node. }\N\end{align*}\N\NThese 17 types are summarized in Table 1, which provides additional details and references to the Sarkisov links associated with each case.