25 problems

Mustată–Srinivas conjecture. There are infinitely many primes $p$ such that

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…

Injective-hull tight-closure conjecture. One has

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 $\mathbb{Q}$-Gorenstein and let $\Delta\geq 0$ be a $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor. Discreteness and rationality conjecture. The set of real numbers $t>0$ suc…

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 $R$ 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 an $n$-dimensional nonsingular variety over an algebraically closed field $k$ of characteristic zero, and let $\mathfrak{a}$ be a nonzero ideal on $X$. For a model $X_A$…

Equality of test ideals conjecture. All versions of test ideals in tight closure theory should be equal.

Let $R=k[x_{11},\dots,x_{mn}]$ be a polynomial ring over a field $k$ of positive characteristic $p$, and let $I=I_t(X_{m\times n})$ be the ideal generated by $t$-minors of…

Let $I$ be an ideal defined over characteristic zero, let $I_p$ denote its reduction modulo a prime $p$, and let $\tau(c\bullet I_p)$ and $\mathcal{J}(c\bullet I)$ denote the corre…

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…

Let $R$ be a ring in the setting above, and write $\tld\tau$ for the big test ideal and $\tau$ for the finitistic test ideal. Big–finitistic test ideal conjecture. It is conjecture…

Let $X$ be a normal $[?]$-Gorenstein closed subvariety of a smooth complex variety $A$. The adjoint-test ideal correspondence conjecture. After reduction to characteristic $p\gg0$,…

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

Finitistic test ideal conjecture. The two test ideals coincide:

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

Let $f\subset\mathbb Q[x_1,\ldots,x_n]$ be an ideal, let $f_p\subset\mathbb F_p[x_1,\ldots,x_n]$ be its reduction modulo $p$, let $\tau(f_p^c)$ be the test ideal, and let…

Let $R$ be a normal ring essentially of finite type over a field of characteristic zero, let $X:=\operatorname{Spec} R$, and let $\Delta$ be an effective $\mathbb{Q}$-divisor on…

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…