33 problems

Let $X=\mathbb{C}P^2\# n\overline{\mathbb{C}P^2}$ carry a symplectic form $\omega$, and let $\Gamma(X,\omega)$ be its Lagrangian root system. Write $PR[\Gamma(X,\omega)]$ for the p…

Let $D_n$ be the circular spherical divisor with $n$ components discussed above. A rational embedding means an embedding of $D_n$ into a rational surface. Rational embeddability co…

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…

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 $N>9$ 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 $S$ 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 $h$ and $p$ be the generators used in the paper for the polynomial expressions $B_{4n}$. For every integer $n$, there is a nonnegative integer $l(n)$ such that…

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…

Let $Z^{\mathrm{sym}}_{N,r}$ denote the symmetric Hilbert modular surface associated with an $(N,r)$-congruence, and suppose $(N,r)$ is one of the pairs in part (i) of the paper's…

Segre–Harbourne–Gimigliano–Hirschowitz conjecture. Either

An $ALG$ or $ALG^*$ gravitational instanton may, after a suitable hyperKähler rotation, admit a compactification obtained by adding a canonical cycle to a rational surface. Compact…

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 $X_m$ be the blowup of $\operatorname{\mathbb{P}}^2$ at $m$ general points. A line bundle $\operatorname{\mathcal{O}}(D)$ is $(-1)$-special if $D$ is effective and…

Let $(X,\omega)$ be a rational symplectic surface with $8\leq\chi(X)\leq 11$, and let $Q(X)=\frac{1}{2}(\chi(X)-2)(\chi(X)-3)$. Let $r^+(X,\omega)$ denote the number of homology cl…

Let $X=\mathbb{CP}^2\#k\overline{\mathbb{CP}}^2$, and let $\omega$ be a generic symplectic form on $X$. Generic connectedness conjecture. The symplectomorphism group…

Let $X= \mathbb{CP}^2koverline{\mathbb{CP}}^2$. Write $pi0\operatorname{Symp}(X)$ for the group of symplectic mapping classes, and call a symplectomorphism a Lagrangian Dehn twist…

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 smooth projective surface. A full exceptional collection in $\mathrm{D}(X)$ is an exceptional collection that generates the derived category; in the line-bundle case,…

Let $X$ be a very general hypersurface of degree $n$ in ${\bf P}^n$, with $n\geq 5$. A hypersurface $X$ is covered by rational surfaces if a family of rational surfaces in $X$ has…

Let $C$ be a prime divisor on ${\rm Bl}_{s}(\mathbb{P}^{2})$. Negative-curve conjecture. If … then $C$ is a $(-1)$-curve. This conjecture describes the expected negative curves on…

Standard-form negativity conjecture. If $s\geq1$ and $D^{2}<0$, then $D$ is not effective. This is presented as a consequence expected from the SHGH conjecture and concerns the eff…