191 problems

Vasconcelos's differential conjecture. If

Let $X\subseteq\mathbb{P}^{4}$ be a threefold hypersurface of degree $d\geq 6$, and let $C\subseteq X$ be any curve. Griffiths–Harris conjecture. One expects … The strongest form o…

Vasconcelos' conjecture. If ${\operatorname{pd}}_R{\rm N}_R<\infty$, then $I$ is generated by a regular sequence.

A complete intersection $X_n(\underline d)\subseteq\mathbb{C}P^{n+k}$ is the transverse intersection of $k$ complex hypersurfaces of multidegree $\underline d=(d_1,\ldots,d_k)$. Le…

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

Moreno's conjecture. The ideal $J$ is almost reverse lexicographic. The conjecture predicts a precise form for the initial ideal of a generic complete intersection. The source late…

Let $\varphi\colon R\to S$ be a homomorphism of noetherian rings essentially of finite type, and let $\mathcal{L}_{\varphi}$ denote its cotangent complex. Write…

Let $R$ be a commutative local Noetherian ring, and let $M$ be a finitely generated $R$-module whose Betti numbers are bounded. Eisenbud's periodicity conjecture. The minimal proje…

Multiplicativity conjecture. Measures of irrationality for complete intersections in projective space should behave multiplicatively in the degrees $d_1,\ldots,d_c$ of the defining…

Herzog's generalized conjecture. If

Distance-irreducible basis conjecture. If $M$ is not distance reducing, then

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

Watanabe's Bold Conjecture. There is another standard graded Artinian complete intersection $B$ of the same socle degree $d$, cut out by quadrics, together with an injective algebr…

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

Singular complete-intersection Knörrer periodicity conjecture. Under appropriate hypotheses on $f_1,\dots,f_k$, there is a quasiequivalence of $A_\infty$-categories

Quillen's conjecture. If $\operatorname{L}(\varphi)$ has finite flat dimension, then $\varphi$ is locally complete intersection.

Vasconcelos's conjecture. If $\Omega_{S/R}$ has finite projective dimension over $S$, then $S$ is a locally complete intersection ring.

Let $A=k[x_0,\ldots,x_m]/I$ be a graded artinian complete intersection $k$-algebra, where $I$ is generated by a regular sequence, and let $L$ be a general linear form. The Strong L…

Cynk–Rams conjecture. If $X$ is CR-nondegenerate in codimension three, then either

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…

Let $X:=V(f_{1},\ldots,f_{k})\subset\mathbb P_{k+3}$ be a nodal complete intersection threefold such that the sequence $(\deg(f_{1}),\dots,\deg(f_{k}))$ is non-decreasing, and supp…

Let $V=V^n(a_1,\dots,a_r)$ be a smooth complete intersection of complex dimension $n$ in $\mathbb{C}P^{n+r}$, defined by hypersurfaces of degrees $a_1,\dots,a_r$, and let $g$ be a…

Let $X\subset \mathbb{P}^5$ be a complex projective complete intersection threefold with isolated singularities, obtained as a complete intersection on a smooth fourfold…