65 problems

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 $S$ be a rational surface, let $D$ be an ample, non-special divisor on $S$, and let $p_1,\ldots,p_r$ be $r$ points on $S$. Write … for the corresponding linear system, and call…

Let $X$ be a rational surface. A standard augmentation is a toric system obtained from a toric system on a Hirzebruch surface by successive blow-ups and the corresponding augmentat…

Let $X=X_{2,s}$ be the blow-up of the projective plane at $s$ general points, and let $D\text{ in }\operatorname{Pic}(X)$ be an effective divisor of the specified form. Write…

Let $X$ be a rational surface, let $v_0$ denote the relevant charge, let $Z_{v_0}(\tau)$ be its partition function, let $\widetilde{Z}_{v_0}^{int}(\tau)$ be the corresponding integ…

Let $X$ be a smooth projective rational surface, and let $n$ be a natural integer. The group of algebraic diffeomorphisms $\operatorname{Diff}_{\mathrm{alg}}(X(\mathbb{R}))$ consis…

Gizatullin's conjecture. If $f^*$ has infinite order on $Pic({\cal X})$, then ${\cal X}$ should have an anti-canonical curve.

Let $n<9$, and consider the rational surface $\mathbb{CP}^2 \# n\,\overline{\mathbb{CP}}{}^2$. A symplectic torus is an embedded symplectic surface of genus one, and two such tori…

Reformulated generalised Harbourne–Hirschowitz conjecture. The class $H$ is non-special and separates $k$-clusters if

Smoothness conjecture for isolated curves. Every such isolated curve $E$ is smooth.

Generalised Harbourne–Hirschowitz conjecture. Let $k\geq 0$ be an integer. If $H$ is an effective divisor class on $X_r$ with

Harbourne–Hirschowitz conjecture. The effective divisors $H$ on $X_r$ that are non-special are exactly those satisfying $H\mathbin{.}E\geq -1$ for every exceptional curve $E$ on…

Let ${\mathbb X}=\{P_1,\ldots,P_s\}\subseteq {\mathbb P}^2={\mathbb P}^2_k$ be a set of $s$ distinct points over an algebraically closed field $k$ of characteristic $0$, and let…

Let $\Sigma_e$ be a rational ruled surface, let $E$ be the curve with $E^2=-e$, and let $F$ be a fiber of the ruling. Set $t^e_\delta(n,m):=t^{\Sigma_e}_\delta(\mathcal O(nF+mE))$,…

Let $\Sigma_e$ be a rational ruled surface, let $E$ be the curve with $E^2=-e$, and let $F$ be a fiber of the ruling. Set $t^e_\delta(n,m):=t^{\Sigma_e}_\delta(\mathcal O(nF+mE))$.…

Let $S$ be a rational surface with effective anticanonical line bundle, and fix a polarization and Chern classes. Consider the moduli space of stable vector bundles on $S$ with tho…

Let $N>9$ be square-free, and let $d/m$ be the fundamental solution of the Pell equation $d^2-Nm^2=1$. Write $(d;m^{\times N})$ for the vector with first entry $d$ and $N$ further…

Let $d,m>0$, and let … be the divisor class on the blow-up $\pi:V_N\to {\mathbf C}P^2$ at $N\geq 9$ generic points. Nagata's conjecture. The divisor $D$ is ample if and only if…

Batyrev–Manin refined counting conjecture. The numbers $\beta$ and $r$ are determined by the geometry of $T$ in a simple way.

Let $S$ be a rational surface, let $N,d,r$ be the parameters in the wall-crossing formula, let $K^2$ denote $K_S^2$, and let $Q_i(N,d,r,K^2)$ and $P_i(N,d,r,K^2)$ be the polynomial…

Let $X$ be a rational surface, let $[?]$...

Let $X$ be a smooth projective surface. The surface is stably rational if $X\times\mathbb{P}^1$ is rational. Stable rationality conjecture for surfaces. If … is rational, then $X$…

Let $X$ and $Y$ be geometrically rational surfaces, and let nontrivial atoms mean the atoms other than the trivial pieces in their atomic semi-orthogonal decompositions. Birational…

Let $X=\mathbb{P}^1\times\mathbb{P}^1$ and let $L=aC_0+bf$. Let $\delta(a,b)$ denote the length of the region of mixed syzygies. Mixed-syzygy length conjecture. The length is … Equ…

Let $X$ be a rational surface and let $L$ be an ample and base point free line bundle on $X$. Suppose that $|L|$ is a linear system containing curves $C$ of genus $g\geq 1$, and wr…