14 problems

Let $X$ be a smooth, irreducible complex algebraic variety and let $f\in\mathcal{O}_X(X)$ be nonzero. Choose a model $(X_A,f_A)$ over a finitely generated ${\mathbf Z}$-subalgebra…

Hacon–Lamarche–Schwede subadditivity conjecture. Given effective ${\mathbb Q}$-divisors $D$ and $E$ on ${\mathscr X}$, one has

Let $X$ be a regular scheme. Subadditivity conjecture. For divisors $\Delta_1$ and $\Delta_2$ on $X$, one has … This is the mixed-characteristic analogue of subadditivity for multi…

Let $X$ be a normal integral scheme projective over a complete local ring $(R,\mathfrak{m})$ with residual characteristic $p>0$, and let $B$ be a $\mathbb{Q}$-divisor on $X$. Pertu…

Let $X$ be a normal integral scheme of finite type over a complete local ring $(R,\mathfrak{m})$ such that $R/\mathfrak{m}$ has positive characteristic. Let $B\geq 0$ be a…

Let $f\in\mathbb{Q}[x_1,\ldots,x_n]$, and for each prime number $p$ let $f_p\in\mathbb{F}_p[x_1,\ldots,x_n]$ be the polynomial obtained by reducing each coefficient of $f$ modulo…

Let $R$ be an excellent Noetherian domain of characteristic $p>0$. Denote by $\tau_{\rm fg}(R)$ the finitistic test ideal and by $\tau_{\rm b}(R)$ the big test ideal of $R$. Test i…

Let $(A,\fra^t)$ be a pair, where $A$ is the regular ring and $\fra$ is the ideal appearing in the construction of the perfectoid test ideal ${\mathrlap{\mJ}\,\,\uptau}(A,\fra^t)$.…

Let $X$ be a normal variety over a field $k$ of characteristic zero, let $\Delta$ be a $\mathbb{Q}$-divisor such that $K_X+\Delta$ is $\mathbb{Q}$-Cartier, and let $\mathfrak{a}$ b…

Reduction-to-characteristic-$p$ multiplier ideal conjecture. For all sufficiently large primes $p$,

Strong-versus-weak F-regularity conjecture. $R$ is strongly $F$-regular if and only if

Let $Y$ be a smooth, irreducible variety over an algebraically closed field $k$ of characteristic zero, and let $\mathfrak{a}$ be a nonzero ideal on $Y$. Given any model $Y_A$ and…

Let $R$ be a domain smooth over ${\mathbf Z}$ with $R\otimes_{\mathbf Z}{\mathbf Q}\neq 0$, let $f\in R$ be nonzero, and for each prime $p$ let…

Let $(R,\mathfrak{m})$ be a regular local ring essentially of finite type over a perfect field of prime characteristic $p$, and let $I \subseteq R$ be a nonzero unmixed ideal. Let…