30 problems

Embedded resolution. There exists a projective, surjective morphism

Resolution conjecture. There exists such a morphism $\pi$.

Let $\chi$ be a reduced analytic vector field on a real analytic manifold $M$ without boundary, and let $U\subset M$ be relatively compact. Write $L_\chi|_U$ for the line field ind…

Let $f:(U_X,X)\rightarrow (U_Y,Y)$ be a proper, not necessarily birational, morphism between nonsingular toroidal embeddings such that … and such that $f|_{U_X}:U_X\rightarrow U_Y$…

Drinfeld's conjecture. The map $\pi$ is a small resolution of singularities of ${\cal Q}_{\alpha}^D$.

Accurate Resolution Conjecture. For an arbitrary Q-Gorenstein threefold $X$ there exists a resolution of singularities $\pi:Y\longrightarrow X,$ such that for every Q-Cartier divis…

Let $X$ be a reduced scheme of finite type over a field $k$ of characteristic $0$, and let $x\in X$ be a closed point. The ridge invariant of $X$ at $x$ is … Here $C_x(X)$ is the t…

Let $X$ be an equidimensional algebraic variety over an algebraically closed field, and let $X^*$ be its Nash blowup, with the normalized Nash blowup obtained by composing the Nash…

Let $X$ be a pure $d$-dimensional $\mathbb{K}$-analytic set, where $\mathbb{K}$ is $\mathbb{C}$ or $\mathbb{R}$, and let $\mathcal{N}(X)$ denote its Nash transformation. A Nash–Sem…

Let $L_0$ be a field and let $L_1$ be a nontrivial finitely generated extension of $L_0$. Suppose there exists a valuation $v$ on $L_1$ with residue field $L_0$. Weak resolution of…

Embedded resolution with boundary. The triple $(\bar{X},D,\bar{Z})$ admits a resolution by blowups in smooth centers

Let $R$ be a complete equicharacteristic local domain with algebraically closed residue field, and let $u$ be a rational valuation centered in $R$ of rational rank $r<dim R$. Let…

Log uniformization conjecture. Every valuation on a quasi-excellent integral scheme $X$ is log uniformizable.

Desingularization conjecture. Every model $X$ possesses a modification

Let $X$ be a singular projective variety, let $\phi:\widetilde{X}\to X$ be any resolution of singularities, and let $E$ be the exceptional divisor. Let $p$ be an integer such that…

Let $X$ be an algebraic or analytic variety, and let $X^{\mathrm{nc}}$ denote its normal-crossings locus. A smooth blowing-up is a blowing-up with smooth centre, and it is admissib…

Let $X$ be an algebraic or analytic variety. A smooth blowing-up is a blowing-up with smooth centre, and it is admissible when it satisfies the local normal-crossings and Hilbert–S…

Let $X$ be a quasi-compact complex manifold, let $\mathcal{F}$ be a codimension-$1$ foliation on $X$ given by an algebraic $1$-form, and let $Z$ be the union of its separatrices. S…

Let $K$ be an algebraically closed field with a non-archimedean valuation $v\colon K\to\Gamma$, let $X$ be as above, and let $\Delta_n$ and $\Gamma^\times$ denote the domains used…

Let $K$ be an algebraically closed field of characteristic $0$, and let $X$ be a reduced separated closed subscheme of finite type over $K$, of pure dimension $r$. Let $X^*$ be the…

Log-resolution conjecture. There exists such a morphism $f:\tilde{X}\to X$ for which $\tilde{X}$ is a regular scheme and $\tilde{X}_s$ is a normal crossings divisor.

A quasi-excellent scheme is a scheme satisfying Grothendieck's and Dieudonné's quasi-excellence conditions. Grothendieck–Dieudonné conjecture. Resolutions of singularities should e…

Weak factorization conjecture. Every birational map between nonsingular varieties has the required factorization.

Let $X\subset Z$ be a reduced closed subscheme of a quasi-excellent regular Noetherian scheme $Z$. Write $X'$ for the strict transform of $X$ under a proper birational morphism…

Let $X_0$ be a reduced algebraic variety over a field of characteristic zero, or a reduced complex- or real-analytic variety, and let $n=\dim X_0$. A resolution of singularities is…