23 problems

Let $S(m)=\mathbb{F}_q[x_0,\dots,x_m]$, and let $S_d(m)$ be its degree-$d$ graded component. For $m,d\geq 1$ and $1\leq r\leq \binom{m+d}{d}$, let $e_r(d,m)$ be the maximum number…

Huybrechts' projectivity and bigness conjecture.

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…

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…

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…

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…

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…

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…

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.

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

Let $Z\subset \mathbb{P}^n$ be the complete intersection of $c$-general hypersurfaces of sufficiently high degree. Let $E_{k,m}T_Z^*$ denote the invariant jet bundle of order $k$ a…

Let $k$ be a field and let $n$ be a positive integer. A smooth projective variety $X$ of dimension $n$ is endowed with a tower of smooth fibrations … where $\dim X_i=n-i$ for each…

Let $X$ be a smooth projective variety defined over a number field, of dimension $n$, and let $S_a(X)$ denote the set of points of type $S_a(X)$. Let $a\geq N$ be a real number. Fu…

Let $V$ be a reduced $n\times(n+r)$ $F$-matrix and let $\gamma\in\mathcal{A}_{\Gamma}(V)$ be a non-singular chamber. A chamber is bordering if it lies on the boundary of the releva…

Let $X$ be a projective hyperbolic manifold. Kobayashi's conjecture. The canonical bundle $K_X$ is ample. This is a necessary-condition prediction related to the Green–Griffiths–La…

Assume that $r$ divides $d$. Let $X=X_{e,d}$ be the variety of binary $d$-ics that are perfect $\frac{d}{e}$-th powers of binary $e$-ics, and let $I_X$ be its defining ideal. Let…

Let $k$ be an algebraically closed field of characteristic $0$, and let $f_1,\ldots,f_l$ be homogeneous polynomials in $k[x_0,\ldots,x_n]$ satisfying … Let…

Degree-12 equations conjecture. The variety $\mathcal{V}_n$ is cut out by equations of degree $12$.