4 problems

Let $\mathscr M_g$ be the moduli stack of smooth genus $g$ curves, let $\eta$ be its generic point, and let $C_{\eta}$ be the universal curve over $\eta$. Write $Pic_{C_{\eta}}$ fo…

Grothendieck's affine morphism conjecture. The induced map on Picard schemes is affine.

Let $X$ be a smooth projective surface over a field of characteristic $p$, and let $Pic_X$ denote its Picard scheme. Bogomolov inequality is the inequality for Chern classes of sem…

Let $S$ be a surface of general type whose Picard scheme is smooth. A version of the Miyaoka–Yau inequality should hold for $S$. Parshin's conjecture. If the Picard scheme of a giv…