63 problems

Let $S=k[X_0,\dots,X_n]$ be a polynomial ring over a field $k$, let $I\subseteq S$ be an ideal, and let $<$ be a monomial order such that the initial ideal $\operatorname{in}_<(I)$…

Schwede–Smith conjecture. $X$ is of Fano type.

Hara–Watanabe conjecture. $(X,\Delta)$ is of strongly $F$-regular type if and only if $(X,\Delta)$ is of klt type.

Hirose–Watanabe–Yoshida conjecture. One always has

A noetherian commutative ring $R$ is a splinter if it splits off as an $R$-module from every finite extension. The conjecture discussed in the source asserts that characteristic…

Let $X$ be a variety over a field of characteristic $0$. Say that $X$ has dense $F$-injective type if, for a model of $X$ over a finitely generated $\a0\mathbb{Z}$-algebra, its red…

Let $x\in X$ be an $n$-dimensional normal $\mathbb{Q}$-Gorenstein singularity over an algebraically closed field of characteristic zero, and suppose that $x$ is an isolated non-log…

Let $(R,\mathfrak{m},k)$ be an $F$-finite local ring of prime characteristic $p>0$. An ideal of $R$ is tightly closed if it equals its tight closure, and $R$ is strongly $F$-regula…

Hara–Watanabe's conjecture. Divisorially F-regular pairs correspond to plt singularities.

Let $X$ be a nonsingular variety over a field of characteristic zero, and let $Y=\sum_{i=1}^k t_iY_i$ be a formal combination where $t_i>0$ are real numbers and each…

Test ideal conjecture. The test ideal defines precisely the non-F-regular locus in $\operatorname{Spec} R$.

Let $\chi$ be the electroid space equipped with the Frobenius splitting under consideration, and let closed electroid varieties be its closed electroid subvarieties. Cohen–Macaulay…

Let $\pi \colon X \to Y$ be a morphism of normal varieties satisfying … Assume that the relative canonical divisor is trivial and that $X$ is quasi-$F$-split. Fiberwise quasi-$F$-s…

Let $X$ be a $3\times 4$ generic matrix of indeterminates, and let $\Bbbk$ be a field of characteristic $p>2$. F-purity conjecture for the $3\times4$ third permanental ideal. The q…

Let $X$ be an $m\times n$ generic, symmetric, or Hankel matrix, and let $\Bbbk$ be a field of characteristic $p\neq 2$. Write $P_t(X)$ for the ideal generated by the $t\times t$ pe…

Hara–Watanabe–Yoshida conjecture. If $T$ is $F$-rational, then $S$ is $F$-rational.

Let $R=\left(\mathbb{C}[X_1, \dots, X_n]/(f_1, \dots, f_r)\right)_{(X_1, \dots, X_n)}$, where each $f_i\in\mathbb{Z}[X_1,\dots,X_n]$, and let $R\bmod p$ denote its reduction modulo…

Let $R$ be a complete local Noetherian ring of characteristic $p$. Let $\Lambda$ be a $p$-base of its coefficient field, and let $R^\Gamma$ denote the ring obtained by the Gamma-co…

Let $(X,B)$ be a normal projective complex pair with klt singularities such that $-K_X-B$ is semiample. K-global F-regularity conjecture. For every model…

Anticanonical bigness conjecture. If $X$ is a splinter, then $-K_X$ is big.

Let $X$ be a globally $F$-split normal projective variety over an algebraically closed field $k$ of characteristic $p>0$. The variety $X$ is globally $F$-split if its Frobenius mor…

Let $X$ be a smooth Calabi–Yau variety over an algebraically closed field $k$ of characteristic $p>0$. The variety $X$ is globally $F$-split if its Frobenius morphism admits a spli…

Deformation conjecture for $F$-purity. Then $R$ is $F$-pure.

Equality of tight closures in injective hulls. One should have

Let $K$ be a field of positive prime characteristic, and let $R$, $\mathcal{I}$, and $J$ be the ring and ideals defined from the commutator matrix in the paper. F-regularity conjec…