36 problems

Let $\mathcal{X}_{\mathrm{cf}}(3,6)$ be the closure of the variety parametrized by the canonical forms of the real regions of $\mathcal{X}(3,6)$, and let the log canonical compacti…

Let $X$ be a smooth projective variety of dimension $n\geq 2$, let $K_X$ be its canonical line bundle, and let $L$ be an ample line bundle on $X$. Moraga–Yeong's conjecture. The co…

Let $d,m$ be positive integers, let $S_d$ be the space of homogeneous degree-$d$ polynomials in $m+1$ variables over ${\mathbb F}_q$, and let $e_r(d,m)$ be the maximum number of…

Let $X$ be a smooth complex projective variety that is the image of a projective toric variety under a morphism. Occhetta–Wiśniewski conjecture. The variety $X$ is automatically to…

Let $X\subset \mathbb{P}^{N}$ be a smooth projective variety, and call a vector bundle on $X$ Ulrich when it satisfies the Ulrich conditions for the given embedding. Ulrich's conje…

Huybrechts' projectivity and bigness conjecture.

Let $X$ be a smooth complex projective variety of dimension $g$. A Hodge class of degree $2d$ is an element of … The cohomology class of an algebraic subvariety of codimension $d$…

Let $X$ be a projective algebraic variety of dimension $n$, let $W$ be a very ample linear system on $X$, and let $S_1,\dots,S_r$ be singularity types with Milnor numbers…

Let $X\subset\mathbb P^n$ be a projective variety of dimension $d\geq 1$, with $\operatorname{reg}(X)=m$ and Hilbert polynomial $P_X$. Let $\delta$ satisfy…

Kovács' conjecture. The variety $X$ is either projective space or the $p$-dimensional quadric hypersurface.

For the products $\mathbb P^1\times\mathbb P^{n-1}$ and $\mathbb P^2\times\mathbb P^{n-1}$, consider the corresponding RR discriminants and their nonisotropic parts. Product RR dis…

Fermat conjecture. Fermat holds inside $V$ if and only if $(V\cap Z)(\mathbb Q)$ is trivial. This gives a proposed characterization of the projective varieties for which the genera…

Eisenbud–Schreyer's conjecture. The set ${\mathrm{Ur}}(X)$ should always be nonempty. This conjecture concerns the existence of Ulrich sheaves for every polarized projective scheme…

Let $e_r(d,m)$ be the maximal number of $\mathbb{F}_q$-rational points of a projective algebraic set defined by $r$ linearly independent homogeneous degree-$d$ polynomials in $m+1$…

Let $e_r(d,m)$ denote the maximal number of $\mathbb{F}_q$-rational points of a projective algebraic set defined by $r$ linearly independent homogeneous degree-$d$ polynomials in…

Let $X\subset \mathbb{P}^n$ be an irreducible projective variety of degree $d\geq 2$. For a positive integer $B$, let $N(X,B)$ denote the number of points in…

Let $X\subseteq\mathbb{P}^{M-1}_{\mathbb{Q}}$ be an irreducible projective variety of degree at least two defined over $\mathbb{Q}$. Let $N_X(B)$ be the number of rational points o…

Expected-codimension and irreducibility conjecture. The variety $\mathcal{V}_{X,\mathcal{C}}$ has expected codimension $n$ inside $\mathcal{M}_\mathcal{C}$. It is positive-dimensio…

Let $X$ be a projective variety defined over a number field $K$, and let $f\colon X\longrightarrow X$ be a surjective self-morphism. Cancellation conjecture. There exists a non-neg…

Picard–analytic equivalence conjecture. The manifold $X$ is $b[31m\ellb[0m$-Picard hyperbolic if and only if it is $b[31m\ellb[0m$-analytically hyperbolic.

Equivalence conjecture. In the category of projective manifolds, the intermediate pseudo-analytic hyperbolicity notions are equivalent.

Reichstein–Rogalski–Zhang conjecture. If $X$ admits a wild automorphism, then $X$ is isomorphic to an abelian variety.

Let $n_1,\ldots,n_r$ be positive integers, and let $d_1,\ldots,d_r$ be positive integers. For a projective variety $X$ and a line bundle $\mathscr{L}$, write…

Quadratic generation conjecture. The ideal defining $\mathcal{R}_{d,k,m}$ is generated by quadratic polynomials.

Let $e_r(d,m)$ denote the maximum number of ${\mathbb F}_q$-rational common zeros in ${\mathbb P}^m$ of $r$ linearly independent homogeneous polynomials of degree $d$, and let…