112 problems

Let $Q$ be a stable partition of length $\ell$, let $J_Q$ be a Jordan matrix of type $Q$, and let $\mathcal N_{J_Q}$ be the variety of nilpotent matrices commuting with $J_Q$. For…

Betti-number conjecture. The ideals $I_h$ and $I_e$ have the same Betti numbers if and only if $n=3a$ for some integer $a$.

Let $\varphi\colon R\to S$ be a ring homomorphism, and let $\operatorname{D}_n(S\!\mid\!R,-)$ denote André–Quillen homology with arbitrary coefficients. Quillen's conjectures. Assu…

Ciolan–García-Sánchez–Moree conjecture. A numerical semigroup $S$ is cyclotomic if and only if it is complete intersection.

Murthy's complete intersection conjecture. For every ideal $I$ in $A$,

Complete intersection classification conjecture. Such a subvariety exists if and only if it is one of the following:

Moreno-Socías's conjecture for generic complete intersections. Then $\operatorname{in}(I)$ is almost reverse lexicographic.

Debarre's ampleness conjecture. There is a positive lower bound $d\gg1$, depending on $N$ and $c$, such that whenever $d_1,\dots,d_c\geqslant d$, the cotangent bundle $\Omega_X$ is…

Moore–Piepmeyer–Spiroff–Walker conjecture. If $R$ is a complete intersection of codimension $c\geq 2$, then

Let $X\subset {\mathbb P}^N$ be covered by lines, let $x\in X$ be a general point, and let $\mathcal L_x\subset {\mathbb P}^{n-1}$ be the variety of lines through $x$. Local-to-glo…

Let $X\subset {\mathbb P}^N$ be a smooth projective variety of dimension $n$ and codimension $c$ satisfying the hypotheses denoted by $(*)$ in the source. Fano Hartshorne conjectur…

Link-existence conjecture. For any $i$, $3\leq i\leq d-2$, if the $i$-th link does not exist, so that there is a common factor in the initial degree of the ideal of $Y_{i-1}$, then…

Exact-one failure conjecture. If there is a complete intersection $J\in CI(d,d,d,d)$ that fails WLP by more than one, then there exists a complete intersection $I\in CI(d,d,d,d)$ t…

Let $A\in\mathbb Z^{n+2}$ be a finite set of full dimension containing $0\in\mathbb Z^{n+2}$ and satisfying Assumption. For generic polynomials $f_1,\ldots,f_{n+1}\in\mathbb C^{A}$…

Let $(X,0)$ be a complete intersection surface singularity whose link is an integral homology sphere. Neumann–Wahl's splice-type equations conjecture. The equations defining…

Let $G\subset\operatorname{SL}(r,\mathbb{C})$ be finite, and consider quotient complete-intersection singularities $\mathbb{C}^{r}/G$ in arbitrary dimension. Existence conjecture.…

Complete-intersection semigroup-condition conjecture. If $X$ is a complete intersection, then $\Delta$ satisfies the semigroup condition.

Let $X=\mathbb{P}^N\subset\mathbb{P}^{N+M}$ be a linear subspace of dimension $N\geq 6$. Let $W$ be a Cohen–Macaulay multiple monomial scheme with reduced subscheme…

Let $\mathbb{P}^N$ with $N\geq 6$ be a linear subspace of codimension two in $\mathbb{P}^{N+2}$. Let $M$ denote the integer occurring in the infinitesimal-neighbourhood notation, a…

Let $(X,o)$ be a Gorenstein normal surface singularity whose link $M$ is an integral homology sphere, meaning $H_1(M;{\mathbb Z})=\{0\}$. Let $\Delta$ be its splice diagram. Integr…

Let $\Bbb{F}$ be a field and let $R$ be a simplicial commutative ring that is locally finite and locally of non-zero characteristic. Say that $R$ is locally a complete intersection…

Quillen's conjecture. The following are equivalent: $D_s(\Bbb{F}|R)=0$ for $s\gg 0$; $D_s(\Bbb{F}|R)=0$ for $s\geq 3$; and $R$ is a complete intersection. This conjecture character…

Let $(R,\mathfrak{m})$ be a regular local ring of dimension $n$. Let $I$ be an ideal generated by an $R$-regular sequence of length $k\leq n$, and let $J$ be an ideal generated by…

Three-variable Schur maximal-rank conjecture. For all sufficiently large $d$, the following holds: if $\lambda\in F$, then $s_\lambda$ fails to be a maximal rank element on $A$; if…

Maximal-rank conjecture for $h_d$. The element $h_d$ is a maximal rank element on $A$ if and only if either $d=2$, $d=3$ and $n\not\equiv 1\pmod{3}$, or $d=4$ and $n=3$.