5 problems

Fano-type conjecture. Globally $+$-regular schemes are of Fano type.

Let $\overline{M}_{g,n}$ be the moduli space of genus $g$ curves with $n$ marked points over $\operatorname{Spec} \mathbb{Z}$, and let $L$ be a nef line bundle on it. Mixed-charact…

Let $G$ be the reductive group, $T$ a maximal torus, and $\rho$ the half-sum of positive roots used to define the affine Grassmannian strata. For a dominant coweight…

Let $S=\operatorname{Spec}(\mathcal O_K)$ be the spectrum of a discrete valuation ring with local parameter $\pi$ and residue field $k$, let $X$ be a scheme of finite type over $S$…

Let $X_S\to S$ be a smooth, projective family over an integral, mixed characteristic base of finite type over $\operatorname{Spec}\mathbb{Z}$, with generic point $\eta$. For a clos…