37 problems

Let $(A,\theta)$ be a polarized abelian variety, and let $C\subset A$ be a smooth curve of genus $h$. Write $c$ for the gonality of $C$ and … for its $\theta$-degree. Gonality–genu…

Nagata–Biran–Szemberg conjecture.

Let $X$ be a projective manifold and let $L$ be an ample line bundle on $X$. For a very general point $x\in X$, write … the maximal value of the Seshadri constant of $L$ on $X$. La…

Debarre's conjecture. If

Let $X$ be a smooth algebraic surface, let $L$ be an ample line bundle on $X$ with degree $d$, and let $p_1,\ldots,p_r$ be sufficiently many general points. The Seshadri constant a…

Let $X$ be a smooth projective surface and let $L$ be a nef divisor on $X$. Let $\epsilon(n;X,L)$ denote the relevant multipoint Seshadri constant for $n$ points. Seshadri constant…

Let $X$ be a smooth variety of dimension $n$ and let $L$ be an ample divisor on $X$. For very general points $p_1,\dots,p_r\in X$, write…

Let $X$ be the variety under consideration, let $D\in \operatorname{Div}(X)\otimes \mathbb{Q}$ be a divisor class on the boundary of the effective cone, and let $\eta\in X$ be a ge…

Let $(A,\theta)$ be an indecomposable principally polarized abelian variety of dimension $g\geq 4$. The hyperelliptic characterization conjecture. If $\varepsilon(A,\theta)<2$, the…

Fix a model $X_\pi$ of $X$, a point $p\in X_\pi$ whose local ring $\mathcal{O}_{X_\pi,p}$ is regular of dimension $c\le \dim X$, and a regular system of parameters at $p$. Let…

Farnik–Szemberg–Szpond–Tutaj-Gasinska conjecture. One has

Let $X$ be a surface and let $H\in\operatorname{Pic} X$ be an ample divisor. For a positive integer $r$, write $\varepsilon(H,r)$ for the multipoint Seshadri constant at $r$ points…

Uniform moving-Seshadri-constant conjecture. There exist integer-valued functions $c(n)$ and $d(n)$ and a real-valued function $e(n)$, all depending only on $n$, such that

Let $(A,\theta)$ be an indecomposable principally polarized abelian variety of dimension $g\geq 5$. For a Jacobian $\operatorname{Jac}C$ of a hyperelliptic curve $C$ with principal…

Let $X_r$ be the blow-up of $[?]$?

Let $X$ be a smooth projective variety over an algebraically closed field. For a coherent sheaf $E$ on $X$ and a point $p\in X$, write $\varepsilon(E,p)$ for its Seshadri constant…

Seshadri lower-bound conjecture.

Piecewise linearity conjecture. The Seshadri function of $A$ is piecewise linear precisely in these cases.

Exceptional-set conjecture.

Finiteness conjecture. There are only finitely many intersection matrices in Picard number two that yield a piecewise linear Seshadri function.

Ein–Lazarsfeld conjecture. There exists a countable union $\mathcal{B}\varsubsetneq X$ of proper subvarieties such that

Uniqueness conjecture. For every ample $\mathbb Q$-line bundle $L$ on $X$, there exists at most one irreducible curve $C$ that is submaximal for $L$ if and only if either

Let $X$ be a smooth projective toric surface, $D$ an ample torus-invariant divisor, $R=(1,1)$ a general point, and $v\in\mathbb{Z}^2$ a primitive direction. Any admissible torus-in…

Let $(A,L)$ be a $g$-dimensional indecomposable polarized abelian variety, meaning that it is not a product of polarized abelian varieties. Let $(J_C,\Theta_C)$ denote the Jacobian…

Let $X$ be a smooth projective variety of dimension $n$ over an algebraically closed field, and let $x_0\in X$. Write $\operatorname{\varepsilon}(TX;x_0)$ for the Seshadri constant…