186 problems

For every projective variety $X\subseteq\mathbb{P}^N$ over a field, equipped with its given embedding, does there exist a nonzero coherent sheaf $\mathcal{E}$ on $X$ that is Ulrich…

Let $p\colon C\to B$ be a smooth proper family of curves, let $L$ be a line bundle on $C$, and let $\pi_n\colon C^{[n]}\to B$ be the relative Hilbert scheme of length-$n$ subscheme…

For $g\geq2$, $n\geq1$, and $d_1,\ldots,d_n\geq1$ satisfying $|\mathbf d|=3g-3+n$, let $C(\mathbf d)$ denote the primitive normalized Witten intersection number, and write…

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…

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

Let $S=\operatorname{Spec} A$ be a discrete valuation ring with perfect residue field $k$ and fraction field $K$. Let $p:X\to S$ be proper and flat of finite type and relative dime…

Let $\overline{M}_{0,n}$ be the moduli space of stable $n$-pointed rational curves, and let $\delta_S$ denote the class of the boundary divisor indexed by $S$, with…

Generalized Bloch conductor formula. If $p:X\to S$ is an arithmetic $S$-scheme with $S$-proper singular locus $Z$, then

Let $E$ be an elliptic curve, let $\overline{M}_{g,n}$ be the moduli space of stable curves, and let $\mathcal C_g(\gamma_1,\ldots,\gamma_n)$ denote the class defined in the paper…

Erdős–Chvátal simplex conjecture. Every family $\mathcal{F}\subseteq\binom{[n]}{k}$ without an $r$-simplex contains at most

Let $g,n,k$ be as above, let $A=(a_1,\ldots,a_n)$ satisfy $\sum_{i=1}^n a_i=k(2g-2)$, and let $\widetilde A=(a_1+k,\ldots,a_n+k)$. For $k\geq 1$, assume that $A$ is not of the form…

Let $h_{g;\mu_1,\ldots,\mu_n}^{\textnormal{one-part}}$ denote the double Hurwitz number with one ramification profile equal to the one-part partition of size…

Weighted Bounded Negativity Conjecture. There exists a nonnegative integer $b_w\in\mathbb Z$ such that

Let $X$ be an integral projective variety of dimension $n$, let $k$ be a nonnegative integer, and let $\operatorname{mob}$ denote the mobility function on the pseudo-effective cone…

Let $(R,\mathfrak{m})$ be a local ring whose completion $\widehat R$ is a hypersurface in an unramified regular local ring. Assume that $R$ is an even-dimensional isolated singular…

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

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:

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…