254 problems

For every integer $n\ge 3$ and every set of $n$ distinct lines in the real projective plane $\mathbb{RP}^{2}$ that are not all concurrent, there exist at least three points of…

Let $X\subseteq\mathbb P^r$ be a nondegenerate projective variety. Its Castelnuovo–Mumford regularity is denoted by $\operatorname{reg} X$, its degree by $\deg X$, and its codimens…

Let $X$ be a smooth complex projective $n$-dimensional variety, let $\mathcal{F}$ be an ample vector bundle on $X$ that is a subsheaf of $\wedge^pT_X$, and let $p>0$. Assume that…

Let $\mathcal{H}$ be an $n$-dimensional Hilbert space, and let $\mathcal{P}_k(\mathcal{H})$ denote the set of rank-$k$ orthogonal projectors on $\mathcal{H}$. Assume that $n=2k$ an…

Chudnovsky's conjecture.

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…

Demailly's conjecture. For every integer $s\geqslant 1$,

Projective-normality and quadratic-generation conjecture. If $g\geq8$ (equivalently, $q(L_2-2\delta)\geq6$), then this embedding is projectively normal; and if $g\geq10$ (equivalen…

Let $X\subset\mathbb{P}^4$ be a very general hypersurface of degree $d\ge6$, and let $C\subset X$ be a curve. Griffiths–Harris conjecture. The degree of $C$ is divisible by $d$. Th…

Secant-dimension conjecture. For each integer $r\geq 1$, we have

The converse to the surplus condition criterion. One has

Let $\omega_{\mathrm{FS}}$ denote the Fubini–Study form on $\mathbb{CP}^n$. For a closed Lagrangian torus $L\subset(X,\omega)$, call $L$ extremal when…

Let $q$ be a power of $3$, let $G=PGL_2(q)$ act on the lines of $PG(3,q)$, and let $C$ be the twisted cubic. A line is generic if it neither intersects $C$ nor lies in any osculati…

Levinson–Ullery conjecture. For $d\geq 1$, if

Let $X$ be a projective manifold with nef anticanonical bundle $-K_X$. For ample divisors $H_1,\ldots,H_{n-1}$, define generic $(H_1,\ldots,H_{n-1})$-semipositivity of $T_X$ by req…

Let $d\geq 3$ and let $Z$ be a set of $\binom{d+2}{2}-1$ general points in $\mathbb{P}^3$. The $d$-Weddle scheme of $Z$ is the scheme associated to the loci of points from which th…

A convex Poncelet polygon is a polygon in the projective plane whose vertices lie on one conic and whose edges are tangent to another conic; convexity means convexity in an affine…

Let $\mathbb{P}_2(n)$ be the projective space consisting of the subspaces of $\mathbb{F}_2^n$, and let a linear code be a subset of $\mathbb{P}_2(n)$ endowed with the linear struct…

Projective-geometry embedding conjecture. There exists an integer $\ell\geq3$ such that every ideal tangled clutter embeds one of $PG(0,2),\ldots,PG(\ell-1,2)$.

Harbourne–Huneke degree-bound conjecture. For every $r>0$,

Let $C\subset \mathbb{P}^2$ be a plane rational cuspidal curve, meaning a rational plane curve whose singularities are cusps. Orevkov's four-cusp conjecture. The curve $C$ can not…

Let $\mathcal L=\mathcal L_{3,d}(m_1,\dots,m_s)$ be a linear system in $\mathbb P^3$ with general base points $p_1,\dots,p_s$. It is Cremona reduced when … for every…

The LQEL classification conjecture. $X$ is obtained by taking linear sections and/or isomorphic projections of a rational homogeneous manifold in its natural minimal embedding.

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…