25 problems

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 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…

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 $X_r$ 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.

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

Debarre's conjecture. If

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…

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…

Let $X$ be a smooth projective variety of dimension $n$, let $\pi:{\rm Bl}_{r}(X)\rightarrow X$ be the blow-up at $r$ general points $x_{1},\dots,x_{r}$, and let $D$ be a big…

Let $X$ be a projective variety and let $\alpha\in\operatorname{Nef}^k(X)$. Suppose that there exists $\varepsilon>0$ such that … for all $x\in X$. Seshadri criterion. Then…

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…

Szemberg's conjecture. For a very general point $x\in X$,

Let $Y$ be any projective manifold, and let $L$ be any ample line bundle on $Y$. The Seshadri constant of $L$ at a point $y\in Y$, denoted $\epsilon(L,y)$, measures the local posit…

Let $\sigma:X\to\mathbb{CP}^2$ be the blow-up at points $x_1,\ldots,x_k$, with $l$ the Poincaré dual of a line and $e_q$ the Poincaré duals of the exceptional divisors. For…

Pell-equation lower-bound conjecture. One should have