184 problems

Zero-dimensional intersection conjecture. Given $d,m\geq 1$ and $m\leq r\leq \binom{d+m}{m}$, we have

Let $X$ be a smooth complex projective surface, and let $C$ be a reduced curve on $X$. Bounded Negativity Conjecture. For each such $X$, there exists a number $B_X\geq0$, depending…

Let $X$ be a compact complex manifold, let $\mathcal{N}\subset H^{1,1}(X,\mathbb{R})$ be its cone of nef $(1,1)$-classes, let $[\alpha]\in\mathcal{N}$, and let $V\subset X$ be a cl…

Alternation conjecture. The Euler characteristic is alternating:

Faber's Gorenstein conjectures. (D) The intersection pairings

Let $S$ be a smooth projective surface and let $L$ be a line bundle on it. Define … Here $H$ is the divisor corresponding to $L$, $K$ is the canonical divisor, and … Let…

Intersection-dimension conjecture. If $\dim R$ is even, then

Let $X$ be a smooth projective variety. Given a basis $\{\gamma_a\}$ for $H^*(X,\mathbb Q)$, let $x_i,y_i\in H^*(X)$ and let $k\geq 2g-3+r+s$. Using Gathmann's conventions … and ……

Let ${\mathcal M}_{g,n}^{rt}$ be the moduli space of stable $n$-pointed genus-$g$ curves with rational tails, and let … be the forgetful morphism. For positive integers…

Multinomial monotonicity conjecture.

Faber–Pandharipande conjecture.

Let $A^*({\overline{\mathcal{M}}}_{0,2}({\mathbb P}^1,2))$ denote the Chow ring of the moduli space of stable maps, let $L_1$ and $L_2$ be the cotangent line bundles associated wit…

Let $(X,B)$ be a log minimal model with $\dim(X)=4$, and let $C$ be a very general curve in $X$. Four-dimensional numerical non-vanishing conjecture. If $(K+B)\cdot C>0$ for very g…

Let $\mathcal{A}$ be a unimodular arrangement, let $\mathfrak{M}(\mathcal{A})$ be the associated hypertoric variety, and let $R(\mathcal{A})$ be the ring generated by the elements…

For $g\geq 0$, $n\geq 1$, and $(g,n)\neq(0,1),(0,2)$, let $H^g_{(d),\beta}$ be the one-part double Hurwitz number, let $r^g_{(d),\beta}$ be its associated number of simple branch p…

Let ${\mathbb P}{\cal H}_n$ be the projectivized Hurwitz space. Let $\Psi_n,\Delta_n\in H^2({\mathbb P}{\cal H}_n)$, where $\Psi_n$ is the distinguished class used for intersection…

Let ${\mathbb P}{\cal H}_n$ be the projectivized Hurwitz space, and let $C_n$ and $M_n$ be the cohomology classes of the caustic and Maxwell stratum, respectively. Consider the sub…

Gorenstein conjecture. The tautological rings of the filtration of $\overline{M}_{g,n}$ are finite-dimensional Gorenstein algebras.

Toric Residue Mirror Conjecture. The Laurent expansion of $R_P(a)$ at the vertex $v_{\mathcal T}\in\operatorname{Sec}(A)$ corresponding to ${\mathcal T}$ coincides with

Let $(A,\mathfrak{n})$ be a quasi-unmixed local ring with prime ideals $P$ and $Q$ such that $\sqrt{P+Q}=\mathfrak{n}$. ID-2 conjecture. If … and both $A/P$ and $A/Q$ are analytica…

Let $(A,\mathfrak{n})$ be a quasi-unmixed local ring with prime ideals $P$ and $Q$ such that $\sqrt{P+Q}=\mathfrak{n}$. ID-1 conjecture. If … and $A/P$ is analytically unramified,…

Expected intersection-point conjecture. For $d>m$ and $d\ge 3$,

Let $G$ be a group, let $g$ denote the genus of $C$, let $\gt$ be the Cartan subalgebra of $G$, let $\M$ denote the moduli space under consideration, and let $P$ be a Weyl-symmetri…

Frankl–Wang conjecture. Under these hypotheses,

Low-codimension pairing conjecture. For $i\leq 3$, these pairings are perfect. If $i=4$, the pairing has rank $\dim \mathsf{R}^4(\mathcal{M}_{g,n}^{\operatorname{ct}})$. The conjec…