19 problems

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…

Nash's conjecture. The number of exceptional curves in the minimal resolution of $(S,s)$ equals the number of irreducible components of $\mathcal{H}$.

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

Embedded resolution. There exists a projective, surjective morphism

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

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

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…

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.

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…

Functorial determination conjecture. There exist a covariant functor $Y:\mathcal{C}'\to\mathcal{C}$ and a natural transformation $F:Y\to\iota$ such that:

Let $Y$ be a reduced quasi-projective variety over $\mathbb{C}$, and let $\rho:Y'\to Y$ be a resolution of the singularities of $Y$. Let $\mathfrak M^{0^{A\!z^f}_{p}}_r(Y)$ denote…

Let $(X,O)$ be a normal surface singularity, and let … be its minimal resolution, with exceptional divisor $E=\bigcup_{k=1}^r E_k$. For each exceptional component $E_k$, let … and…

Let $(\Sigma_0,\Delta_0,\xi_0)$ be a coordinated normal crossing scheme over $k$, and let $\Phi_0$ be any effective divisor of $\Sigma_0$. A weakly admissible composition of blowin…