3 problems

Ciliberto–Di Gennaro conjecture. The hypersurface $X$ is either factorial, or contains a plane and has at least $(d-1)^2$ nodes, or contains a quadric surface and has exactly…

Let $Y$ be a nodal hypersurface of degree $d$ in $\mathbb{P}^4$ with $s$ singular points. Q-factoriality conjecture. If $s<(d-1)^2$, then $Y$ is $\mathbb{Q}$-factorial; if $Y$ cont…

Let $X$) be a nodal projective threefold, and suppose that each node carries a local torsion package $E_i$ realized in degree-three support cohomology, so that … is defined. Defect…