52 problems

Let $X$ be a surface and let $A$ be an ample line bundle on $X$. For an integer $p\geq 0$ and an integer $\ell$, set $L=K_X+\ell A$. Mukai's conjecture. The line bundle $L$ satisfi…

Let $(p\in X)$ be a cusp singularity. A Looijenga pair is a pair $(Y,D)$ consisting of a smooth rational surface $Y$ and an anticanonical divisor $D$ that is a cycle of rational cu…

Loewner's bound. Every aspherical surface satisfies

Sabbah's conjecture. There is a finite sequence of point blow-ups

Schläfli–Yau conjecture. Every two-dimensional Riemannian manifold has a local isometric immersion into $\mathbb{R}^{3}$.

Relative Pu conjecture. There is a point $p\in X$ such that

Uniform boundedness conjecture. There is a constant $C$ depending only on the genus of $S$, the area of $S$, the minimum length of the geodesics in $S$, and the minimum of the curv…

Let $S$ be a minimal surface of general type with Albanese dimension $2$ and irregularity $q(S)\ge 4$. The q≥4 conjecture. Then … Moreover, equality holds if and only if $S$ is a p…

Let $S$ be a minimal surface of general type with Albanese dimension $2$ and irregularity $q(S)=3$. The q=3 conjecture. Then … and equality holds if and only if $S$ is the symmetri…

Let ${\cal W}$ be the Willmore functional on tori, and consider deformations generated by the first modified Novikov–Veselov (mNV) equation that preserve tori. Stationarity conject…

Let $g\geq 2$, let $S_g$ be the closed orientable surface of genus $g$, and let a simple graph be called rigid in $S_g$ when it has a well-positioned realisation whose associated g…

Resonance-multiplicity conjecture. The following assertions hold:

Let $S$ be the surface in the preceding setup, let $C_i$ be one of the distinguished curves, and let $[\mathfrak Y_{i,(d)}]$ and $[\mathfrak Z_{i,\ell}]$ denote the corresponding d…

Barja–González–Naranjo's modified conjecture. For every non-isotrivial fibred surface $f\colon S\to B$ of genus $g\geq 2$,

Let $(S,V)$ be a marked surface with Euler number $\chi(S)\leq 0$, and let $(dist_S,r)$ be a decorated piecewise Euclidean metric on $(S,V)$. A decorated PE metric is discrete conf…

A surface has the n-circle property if, through every point on it, exactly $n$ distinct circles can be drawn on the surface. Blum's conjecture. There are no surfaces with the $n$-c…

Generalized Bloch conjecture. If $[Z]^*$ vanishes on $H^0(X,\Omega^2_X)$, then $Z_*$ vanishes on $F^2CH_0(X)$.

Taimanov's sphere inequality conjecture. The inequality ${\cal W}\geq 4\pi N^2$ holds for all spheres, not only for spheres with a one-dimensional potential. The text presents this…

Let a torus evolve under the modified Novikov–Veselov (mNV) flow, and regard two surfaces differing by translations as equivalent. A torus is nonstationary if its mNV evolution is…

Let $n$ denote the length of the cycle associated with a symmetric cusp singularity equipped with an antisymplectic involution. Length-12 conjecture. The value $n=12$ is large enou…

Let $(p\in X)$ be a cusp singularity equipped with an antisymplectic involution $\iota$ fixed point free on $X\setminus\{p\}$, and let $D$ be its dual cycle. Let $Y$ be a smooth ra…

Let $(p\in X)$ be a cusp singularity equipped with an antisymplectic involution $\iota$. Let $D$ denote the dual cycle of the cusp. An equivariant smoothing conjecture. Then…

Let $Y$ be the surface, let $D\subset Y$ be the boundary divisor, let $U=Y\setminus D$, and let $Z$ be the union of all $(-2)$ curves in $Y$. Write $Z'=Z\cap U$, let $\bar{Q}$ be t…

Assume that $D$ is negative definite. Let $D_1,\ldots,D_k$ be the irreducible components of $D$, and define … Let $\mathcal{C}$ be the interior of…

Let $Y$ be the surface, let $D\subset Y$ be the boundary divisor, and put $U=Y\setminus D$. Let $\mathcal{A}'\subset\mathcal{A}$ be obtained by restricting to curves contained in…