7 problems

Let $(X,\Delta)$ be a projective log Fano pair over $\mathbb{C}$, and let $(X_p,\Delta_p)$ denote reductions to characteristic $p$ for sufficiently large primes. Let…

Let $(R,\mathfrak{m})$ be a KLT singularity essentially of finite type over $\mathbb{C}$. For every sufficiently large prime $p$, let $\mathscr{s}_p(R)$ denote the $F$-signature of…

Let $X$ be a $d$-dimensional $\mathbb{Q}$-Fano variety over $\mathbb{C}$. Q-Fano F-alpha conjecture. … This is a weaker global version of the positivity conjecture for log Fano pai…

Let $(R,\mathfrak m)$ be a $d$-dimensional Gorenstein local domain essentially of finite type over $\mathbb{C}$. Gorenstein positivity conjecture. If $\operatorname{Spec} R$ has kl…

For $d\geq2$, let … be the degree-$d$ Fermat hypersurface ring, and define $c_n$ and $c'_n$ by … … Here $\xi_{2d}$ is a primitive $2d$-th root of unity. Rationality conjecture. The…

Let $x\in X$ be an $n$-dimensional klt singularity in characteristic $0$, and for primes $p\gg0$ let $x_p\in X_p$ be its reduction modulo $p$. Let $s(x_p,X_p)$ denote the F-signatu…

Let $(A,\mathfrak{m})$ be a non-regular local ring of dimension $d \geq 1$, and let $A_{p,d}$ be the simple singularity defined above. Maximal F-signature conjecture. … Since…