Rationally connected threefolds with nef and bad anticanonical divisor (Q6621670)

Let \(X\) be a smooth complex projective rationally connected threefold. The anticanonical bundle is \(-K_X\) is a pivotal object in classification of \(X\):\N\begin{itemize}\N\item[1.] When \(-K_X\) is ample, Mori and Mukai \((\rho>1)\), and Iskovskish \((\rho=1)\) gave complete classfication of \(X\), according to the Picard rank \(\rho\).\N\item[2.] When \(-K_X\) is nef but not numerically trivial, Birka, Di-Cerbo and Svaldi showed such threefolds are birationally bounded.\N\item[3.] We say a nef divisor \(D\) is good, if the Iitaka dimension \(\kappa(D)\) equals the numerical dimension \(\nu(D)\). Otherwise we say the nef divisor \(D\) is bad.\N\begin{itemize}\N\item[(a)] For varieties with good \(-K_X\), then \(-K_X\) is semi-ample, and standard methods in MMP classifies \(X\).\N\item[(b)] For vareities with bad \(-K_X\), Bauer and Peternell showed that the nef-dimension \(n(-K_X)=3\), the numerical dimension \(\nu(-K_X)=2\) and Iitaka dimension \(\kappa(-K_X)=1\). The result naturally leads to studies of base locus of the \(-K_X\).\N\end{itemize}\N\end{itemize}\N\NThe paper under review extended the discussion in (3b), and showed the following results\N\begin{itemize}\N\item[1.] If \(-K_X\) has no fixed part, then \(-K_X\sim 2D\) and \(X\) can be explicitly classified.\N\NThe result deduces the boundedness of smooth complex projective rationally connected threefolds with nef (and not semi-ample) \(-K_X\) when the \(|-K_X|\) has no fixed part.\N\item[2.] If \(-K_X\) has a non-empty fixed part, then after a sequence of flops, one can always assume the mobile part is nef.\N\NThis result does not give boundedness, however the author expects a complete classification as in previous case. The author refers further discussion to his thesis [\textit{Z. Xie}, Géométrie des diviseurs anticanoniques pour certaines variétés rationnelle ment connexes de petite dimension. Université Côte d'Azur (PhD Thesis) (2021)].\N\end{itemize}