520 problems

Eisenbud–Goto conjecture for selected classes. The conjecture may hold for smooth projective varieties, projectively normal varieties, projective toric varieties, and projective si…

Let $X$ be a smooth subvariety of $\mathbb{P}^n$ over $\mathbb{C}$. An Ulrich bundle on $X$ is a nonzero vector bundle satisfying the equivalent linear-resolution, cohomological-va…

Let $I$ be the defining ideal of $s$ reduced points in ${\mathbb P}^N$, let $\mathfrak m$ denote the homogeneous maximal ideal, and let $h$ be the parameter appearing in the contai…

Let $f:S\to\mathbb P^2$ be a generic cover, with branch curve $B_f$. Two generic covers $f:S_i\to\mathbb P^2$, $i=1,2$, are equivalent when they have the same branch curve and are…

Let $\mathrm{PG}(3,q)$ be three-dimensional projective space over $\mathbb{F}_q$. A Cameron–Liebler line class is a set of lines meeting every spread in a fixed number $x$ of lines…

Laface–Ugaglia conjecture. The special effect varieties for Cremona reduced linear systems in $\mathbb{P}^3$ are lines through two points and quadrics through nine points.

Let $d\geq 2$ and let $Z$ be a set of $\binom{d+2}{2}-1$ general points in $\mathbb{P}^3$. The $d$-Weddle scheme is the locus of points $P$ for which the projection $\pi_P(Z)$ lies…

Let ${\mathbb{X}}\subseteq {\mathbb{P}}^n$ be a set of points having $\mathop{\rm CBP}\nolimits(r)$. A plane configuration $\mathcal{P}$ is a union of distinct positive-dimensional…

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…

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

Let $\mathrm{Seg}(\mathbb P^2\times\mathbb P^2)\subset\mathbb P^8$ be the Segre variety. Griffiths–Harris rigidity conjecture. The Segre variety can be characterized by a second-or…

Let the Hilbert scheme of surfaces in $\mathbb{P}^4$ parametrize surfaces in projective four-space, and consider its irreducible components whose general members correspond to smoo…

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 $\mathcal{F}$ be a two-dimensional holomorphic foliation on $\mathbb{P}^{3}$. Brunella's conjecture. The foliation $\mathcal{F}$ either admits an invariant algebraic surface or…

Irreducibility conjecture. For each $n\ge 3$, there exists a value of $c$ such that $C_f$ is irreducible.

Let $q\geq 5$ be a prime power, with the corresponding parity and residue value $\xi\in\{-1,0,1\}$ determined by $q\equiv\xi\pmod 3$. Let $G_q$ act on the EnΓ-lines in…

Tsfasman's conjecture. If $d\leq q+1$, then

Gross–Popescu's conjecture. The homogeneous ideal of $A$ in this embedding is generated by quadrics and cubics.

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 $\gamma\subset\mathbb{RP}^2$ be a strictly convex planar curve equipped with a family $(\sigma_P)_{P\in\gamma}$ of projective involutions of the tangent lines $L_P$, each fixin…

Let $M$ be a compact complex manifold. An algebraic foliation is a foliation on a projective variety defined in the algebraic category, and a submanifold is transverse to the folia…

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…

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 $X\subset\mathbb{C}\mathbb{P}^n$ be a closed irreducible subvariety with $X^\circ=X\setminus\mathcal{H}\neq\emptyset$. Its $i$-th sectional maximum likelihood degree is … where…

Iarrobino's conjecture. For every $n\in\mathbb{N}$,