28 problems

Let the DSII hierarchy be the integrable hierarchy governing deformations of surfaces immersed in $\mathbb{R}^4$, $\mathbb{R}^{3,1}$, and $\mathbb{R}^{2,2}$ via generalized Weierst…

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

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

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 $g\geq 2$, 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…

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 $S$ 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 $g\geq 2$,

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

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 under consideration, and let $i:C\hookrightarrow Y$ denote the inclusion of a $(-2)$ curve. Define the collection of spherical objects … Let…

Let $S$ be a simply-connected surface with Hodge numbers $h^{1,0}=h^{2,0}=0$. A geometrically maximal degeneration is a degeneration having the geometric maximality property used i…

Let $X^{\varepsilon}$ be the generic fiber of the degeneration constructed in the paper, and let a deformation class mean an equivalence class of surfaces connected by a deformatio…

Let $g$ denote the genus and let $\boldsymbol{\iota}_0\in(0,1]$ be the reduced area difference parameter for the Canham model. Let … A solution is considered up to homothetic trans…

Zariski's multivariable Poincaré-series conjecture. The series $M_{X,D,D_1,D_2,\ldots,D_l}(t_1,t_2,\ldots,t_l)$ is rational: there exist $f,g\in{\mathbb Z}[t_1,t_2,\ldots,t_l]$ wit…

Let $I_k(\gamma)$ denote the $k$-th Friedlander–Nadirashvili invariant for a non-orientable surface of genus $\gamma$, and suppose that $\gamma$ is even. Even-genus conjecture. For…

Let $X$ be a smooth projective surface, either rational or complex, and let $C$ be an effective reduced irreducible divisor on $X$. Irreducible bounded negativity conjecture. There…

Let $S$ be a smooth projective surface, and let $\widetilde{S}$ be its universal covering space. Infinite-chain conjecture. The infinite chain condition can be ruled out for…

Finite-order automorphism conjecture. The induced map

Let $S\subset\mathbf{P}^3$ be a general surface of degree $d\geq 4$. Let $V_2(S,\mathcal{O}_S(1))$ be the curve of binodal plane sections of $S$, and let…

Loewner's bound. Every aspherical surface satisfies