42 problems

Let $X$ be a smooth projective variety over $\bC$, let $A=\operatorname{Spec}R$ where $R$ is a finitely generated $\bZ$-algebra containing $\bZ$, and let $X_A\to A$ be a spreading…

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…

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…

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

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.

Hirose–Watanabe–Yoshida conjecture. One always has

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…

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

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…

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

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…

Let $(A,\mathfrak{m}, k)$ be a formally unmixed non-regular local ring of dimension $d \geq 1$ with algebraically closed residue field. Let $m \geq 1$ be an integer, and let…

Let $(R,\mathfrak m)$ be an F-rational local ring. Write $\operatorname{s_{rel}}(R)$ for its relative F-rational signature and $\operatorname{s_{dual}}(R)$ for its dual F-rational…

Let $X$ be a normal projective variety over an algebraically closed field $k$ of characteristic zero. An endomorphism $f\colon X\to X$ is int-amplified if there exists an ample Car…

F-rational fiber-locus conjecture. The locus of $t\in\mathfrak{mSpec}D$ for which $X_t$ is $F$-rational is constructible. If the family is flat and proper, or sufficiently local, t…

Let $x\in (X,\Delta)$ be a klt singularity over characteristic $0$, and let $(x_p\in (X_p,\Delta_p))$ be its reduction modulo $p\gg 0$. Reduction conjecture. … This conjecture prop…

Let $(R,\mathfrak{m},k)$ be a local or graded ring of prime characteristic $p>0$, and let $I\subseteq R$ be an ideal, homogeneous in the graded case. Write $I^{[p^e]}$ for the idea…

F-regularity conjecture for partial commutator ideals. The quotient $R/\mathcal{I}_Z$ is $F$-regular. In particular, $\mathcal{I}_Z$ is a prime ideal.

F-regularity conjecture for $\mathcal{P}(X)$. The quotient ring

F-purity conjecture. The quotient ring $R/I$ is $F$-pure for all $n$.

F-regularity conjecture for nearly commuting matrices. The rings