17 problems

Let $X$ be a rational surface and let $\mathcal{A}$ be a toric system on $X$. Denote by $\mathbb{TV}(\mathcal{A})$ the smooth toric variety associated with $\mathcal{A}$. Bondal's…

Positivity and monotonicity conjecture. The Welschinger invariants $W_{D,m}$ are positive if $m<[(c_1(\Sigma)\cdot D-1)/2]$, non-negative if $m=[(c_1(\Sigma)\cdot D-1)/2]$, and sat…

Let $\mu$ be the discrete parameter, let $\mathfrak p^{(I)}$ denote the collection of equivariant flux data, and let $\epsilon_1,\epsilon_2$ be the equivariant parameters. Reflecti…

Let $S$ be a toric surface, and consider the relevant linear systems and their universal curves. Let the associated relative Hilbert schemes of points on these universal curves car…

In the toric surface setting, fix a genus and consider the coefficients of fixed codegree in the refined invariants as functions of the degree; these coefficients are ultimately po…

Sharpness conjecture. If $X$ is $\mathbb P^2$, $\mathbb P^1\times\mathbb P^1$, or a Hirzebruch surface, then

Let $n\geq 3$ and let $\mathbf{a}=(a_0,\ldots,a_{n-1})$ define a Fulton toric cycle, meaning that … A lifting of the Fulton cycle $A(\mathbf{a})$ is a nice cycle…

Triangular conjecture. All matrices $A_{ij}(\mathbf{a})$ are elementary.

Let $X$ be a smooth complete toric surface, let $\mathcal{L}$ be a line bundle in the setting of the monodromy map, and let $C$ be a curve whose nodes form the set on which monodro…

Batyrev–Manin–Peyre conjecture. There exists an open subset $U\subseteq X$ such that

Let $X$ be a smooth toric surface and let $\mathcal L$ be an ample line bundle on $X$ whose generic fiber is not hyperelliptic. Assume $r>1$ or $g(\mathcal L)\geq 5$, and let…

Let $X$ be a smooth and complete toric surface, let $\mathcal{L}$ be an ample line bundle on $X$ such that the curves in $|\mathcal{L}|$ have arithmetic genus at least $1$, and let…

Generating-function conjecture. This formula is conjectured to be the generating function of equivariant Donaldson invariants with $c_1=0$ on $\mathbb{P}^2$.

Let $\Delta$ be a lattice polygon, let $g$ be its number of interior lattice points, and let $S$ be a set of $n+1-g$ points in the dense torus of the associated polarized toric sur…

Let $S$ be a toric Fano surface, let $\beta$ be a curve class of degree $d$, and let $N^{g(\beta)-\delta}_\beta(\kappa_S)$ denote the transformed Gopakumar–Vafa invariant, where…

Toric-surface exceptional-sequence conjecture. Either $\mathcal{E}$ is constructible, or it differs from a constructible exceptional sequence by a finite sequence of spherical twis…

Flenner–Kaliman–Zaidenberg conjecture. Among smooth affine $\mathbb{C}^*$-surfaces, the toric surfaces and the Danilov–Gizatullin surfaces are the only exceptions to uniqueness of…