112 problems

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…

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

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 $n=3$.

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

Let $R$ be a complete intersection ring with $k$ relations and $n$ variables, where $k\geq 1$. For any map $f:\mathbb{N}\longrightarrow\mathbb{N}$, define $\Delta^0f(m)=f(m)$ and…

Assume that a local ring $R$ has minimal regular presentation $Q/I$, let $k$ be its residue field, and fix minimal generators $f_1,\ldots,f_n$ of $I$. Let … and let … where each…

Let $S=\mathbb{C}[x_1,\ldots,x_n]$ with $n\geq 3$, and let $f_j\in S_{d_j}$ be homogeneous polynomials of degrees $d_j\geq 2$ such that … is a zero-dimensional complete intersectio…

Herzog's generalized conjecture. If

Herzog's cotangent homology conjecture. The following conditions are equivalent:

Regularizing fractional-transformation conjecture. All such Calabi–Yau Laurent hypersurfaces are also describable, via suitable fractional transformations, in terms of general-type…

Herzog's last theorem. The following conditions are equivalent:

Herzog's conormal module conjecture. If $I/I^2$ has finite projective dimension as an $R/I$-module, then $I$ is a complete intersection.

A toric variety is an algebraic variety equipped with the torus action arising from a toric construction; a Calabi–Yau hypersurface is a hypersurface with trivial canonical class.…