126 problems

Donald's conjecture. One has

Delta-invariant conjecture. For every $n$, one has

Batyrev–Popov conjecture. Cox rings of del Pezzo surfaces are presented by ideals of quadrics.

Let $S$ be a log del Pezzo surface, meaning a del Pezzo surface with at worst klt singularities. Let … be its cylindrical ample set, where $\mathrm{Amp}(S)$ is the ample cone. Chel…

Picard number conjecture. Then $S$ has Picard number one.

Assume that $X$ is rational, $D$ is a smooth anticanonical curve on $X$, and $P\in D(\beta)$, where $D(\beta)$ is the finite set of points associated with the class $\beta$. Let…

Relative BPS integrality conjecture. For every $d\geq 1$,

Let $(X_i,\omega_i)$ be a sequence of smooth Kähler–Einstein Del Pezzo surfaces converging in the Gromov–Hausdorff sense to a limiting metric space $X$. Tian's conjecture. The limi…

Batyrev's conjecture. The universal torsor over any del Pezzo surface admits an embedding into the homogeneous space defined by the orbit of the highest weight vector of the repres…

Let $G_r$ be the simply connected semisimple group of type $R_r$, let $H_r\subset G_r$ be a fixed maximal torus, and let $P_r\subset G_r$ be the maximal parabolic subgroup defined…

Strong rational connectedness conjecture. The smooth locus $X^{sm}$ of a log Del Pezzo surface $(X,\Delta)$ is strongly rationally connected.

Let $S\subseteq\mathbb{P}^d$ be a possibly singular del Pezzo surface, let $U\subseteq S$ be obtained by deleting exceptional divisors, and let $\rho_S$ be the relevant Picard rank…

Let $Q$ be a root of unity of order $k$ prime to $l$. Consider the canonical equation associated with the corresponding quantized del Pezzo surface, whose coefficients are function…

Asymptotic conjecture. For every fixed $r”\ge 0$,

Positivity and monotonicity conjecture. The invariants are positive for

Generalized Eckardt point conjecture. If the total log canonical threshold of $X$ is $\frac{n-1}{n}$, then a wild tiger $S$ for $X$ is a cone in $\mathbb{P}^{n-1}$ over a smooth hy…

The Chern–Simons system considered in this section has a free energy whose perturbative expansion is compared with the open-string amplitudes, with the local $dP_2$ model denoting…

Let $X$ be either the Hirzebruch surface $\mathbb F_2$ or the surface $B'$ obtained by blowing up six points on a conic, and let $E$ be its $(-2)$-curve. Let $D$ be an effective di…

Let $X$ be a model over the local ring $S=\operatorname{Spec}{\mathcal O}$ of a smooth complex curve, with generic fiber a Del Pezzo surface. Assume that $X$ has terminal singulari…

McDuff–Siegel's conjecture. The same correspondence between rigid unicuspidal symplectic curves and rigid unicuspidal algebraic curves should hold for other del Pezzo surfaces, inc…

Let $Q_{\operatorname{/}}$ be the reduced quiver associated with a well-split configuration, and let $\Sigma$ be a polygon bounded by arrows of $Q_{\operatorname{/}}$. Herzog's pol…

Let $S$ be a quasi-smooth, well-formed del Pezzo hypersurface of degree $d$ in the weighted projective space … where $S$ is defined over an algebraically closed field $\Bbbk$ of ch…

Let $M$ be a del Pezzo surface of degree $1$, $5$, or $7$, and let $D$ be a generic smooth representative of $M$'s anticanonical divisor. A stable $\mathbb{R}$-polarization of $X$…

Let $S$ be a log del Pezzo surface. The cylindrical ample cone conjecture. If the cylindrical ample cone … contains the anticanonical class, then … Equivalently, the cylindrical am…

Quartic near-miss infinitude conjecture. If this equation has an integer solution, then it has infinitely many integer solutions. The conjecture is suggested by the paper's non-thi…