We prove that the algorithm for desingularization of algebraic varieties in characteristic zero of the first two authors is functorial with respect to regular morphisms. For this purpose, we show that, in characteristic zero, a regular morphism with connected affine source can be factored into a smooth morphism, a ground-field extension and a generic-fibre embedding. Every variety of characteristic zero admits a regular morphism to a Q-variety. The desingularization algorithm is therefore Q-universal or absolute in the sense that it is induced from its restriction to varieties over Q. As a consequence, for example, the algorithm extends functorially to localizations and Henselizations of varieties.
In characteristic zero, we construct relative principalization of ideals for logarithmically regular morphisms of logarithmic schemes, and use it to construct logarithmically regular desingularization of morphisms. These constructions are relatively canonical and even functorial with respect to logarithmically regular morphisms and arbitrary base changes. Relative canonicity means, that the principalization requires a fine enough non-canonical modification of the base, and once it is chosen the process is canonical. As a consequence we deduce the semistable reduction theorem over arbitrary val…
We prove that any noetherian quasi-excellent scheme of characteristic zero admits a strong desingularization which is functorial with respect to all regular morphisms. We show that as an easy formal consequence of this result one obtains strong functorial desingularization for many other spaces of characteristic zero including quasi-excellent stacks and formal schemes, and complex and non-archimedean analytic spaces. Moreover, these functors easily generalize to non-compact setting by use of converging blow up hyper-sequences with regular centers.
We introduce the notion of combinatorial type of varieties $X$ which generalizes the concept of the dual complex of SNC divisors. It is a unique, up to homotopy, finite simplicial complex $Σ(X)$ which is functorial with respect to morphisms of varieties. Its cohomology $H^i(Σ(X),Q)$ for complex projective varieties coincide with weight zero part of the Deligne filtration $W_0(H^i(X,Q))$. The notion can be understood as a topological measure of the singularities of algebaric schemes of finite type. We also prove that any variety in characteristic zero admits the Hironaka desingularization with …
Let $X$ be any variety in characteristic zero. Let $V \subset X$ be an open subset that has toroidal singularities. We show the existence of a canonical desingularization of $X$ except for V. It is a morphism $f: Y \to X$ , which does not modify the subset $ V $ and transforms $X$ into a toroidal embedding $Y$, with singularities extending those on $V$. Moreover, the exceptional divisor has simple normal crossings on $Y$. The theorem naturally generalizes the Hironaka canonical desingularization. It does not modify the nonsingular locus $V$ and transforms $X$ into a nonsingular variety $Y$. Th…
We present a complete generalization of Kirwan's partial desingularization theorem on quotients of smooth varieties. Precisely, we prove that if $\mathcal{X}$ is an irreducible Artin stack with stable good moduli space $\mathcal{X} \to X$, then there is a canonical sequence of birational morphisms of Artin stacks $\mathcal{X}_n \to \mathcal{X}_{n-1} \to \ldots \to \mathcal{X}_0 = \mathcal{X}$ with the following properties: (1) the maximum dimension of a stabilizer of a point of $\mathcal{X}_{k+1}$ is strictly smaller than the maximum dimension of a stabilizer of $\mathcal{X}_k$ and the final s…
Algorithms for resolution of singularities in characteristic zero are based on Hironaka's idea of reducing the problem to a simpler question of desingularization of an "idealistic exponent" (or "marked ideal"). How can we determine whether two marked ideals are equisingular in the sense that they can be resolved by the same blowing-up sequences? We show there is a desingularization functor defined on the category of equivalence classes of marked ideals and smooth morphisms, where marked ideals are "equivalent" if they have the same sequences of "test transformations". Functoriality in this sen…
Let $n\geq 4$, $2 \leq r \leq n-2$ and $e \geq 1$. We show that the intersection of the locus of degree $e$ morphisms from $\mathbb{P}^1$ to $G(r,n)$ with the restricted universal sub-bundles having a given splitting type and the locus of degree $e$ morphisms with the restricted universal quotient-bundle having a given splitting type is non-empty and generically transverse. As a consequence, we get that the locus of degree $e$ morphisms from $\mathbb{P}^1$ to $G(r,n)$ with the restricted tangent bundle having a given splitting type need not always be irreducible.
Following Mumford and Chiodo, we compute the Chern character of the derived pushforward $\textrm{ch} (R^\bulletπ_\ast\mathscr{O}(\mathsf{D}))$, for $\mathsf D$ an arbitrary element of the Picard group of the universal curve over the moduli stack of stable marked curves. This allows us to express the pullback of universal Brill-Noether classes via Abel-Jacobi sections to the compactified universal Jacobians, for all compactifications such that the section is a well-defined morphism.
To any finite local embedding of Deligne--Mumford stacks $g: Y\to X$ we associate an étale, universally closed morphism $F_{Y/X}\to X$ such that for the complement $Y^2_X$ of the image of the diagonal $Y \to Y\times_XY$, the stack $F_{Y^2_X/Y}$ admits a canonical closed embedding in $F_{Y/X}$, and $F_{Y/X}\times_XY$ is a disjoint union of copies of $F_{Y^2_X/Y}$. The stack $F_{Y/X}$ has a natural functorial presentation, and the morphism $F_{Y/X}\to X$ commutes with base-change. The image of $Y^2_X$ in $Y$ is the locus of points where the morphism $Y \to g(Y)$ is not smooth. Thus for many prac…
We show that every unramified morphism X->Y has a canonical and universal factorization X->E->Y where the first morphism is a closed embedding and the second is etale (but not separated).
We show that a toroidal morphism can be reduced to a weakly semistable one in a universal way if we allow families to be modified to Deligne-Mumford stacks instead of schemes.
We expose in detail the principle that the relative geometric invariant theory of equivariant morphisms is related to the GIT for linearizations near the boundary of the $G$-effective ample cone. We then apply this principle to construct and reconstruct various universal moduli spaces. In particular, we constructed the universal moduli space over $\overline{M_g}$ of Simpson's $p$-semistable coherent sheaves and a canonical dominating morphism from the universal Hilbert scheme over $\overline{M_g}$ to a compactified universal Picard.
Let G be a unitary group over a totally real field, and X a Shimura variety for G. For certain primes p of good reduction for X, we construct cycles on the characteristic p fiber of X. These cycles are defined as the loci on which the Verschiebung map has small rank on particular pieces of the Lie algebra of the universal abelian variety on X. The geometry of these cycles turns out to be closely related to Shimura varieties for a different unitary group G', which is isomorphic to G at finite places but not isomorphic to G at archimedean places. More precisely, each such cycle has a natural des…
We introduce the notion of families of n-marked smooth rational tropical curves over smooth tropical varieties and establish a one-to-one correspondence between (equivalence classes of) these families and morphisms from smooth tropical varieties into the moduli space of n-marked abstract rational tropical curves.
We show that deformations of a surjective morphism onto a Fano manifold of Picard number 1 are unobstructed and rigid modulo the automorphisms of the target, if the variety of minimal rational tangents of the Fano manifold is non-linear or finite. The condition on the variety of minimal rational tangents holds for practically all known examples of Fano manifolds of Picard number 1, except the projective space. When the variety of minimal rational tangents is non-linear, the proof is based on an earlier result of N. Mok and the author on the birationality of the tangent map. When the varieties …
In their book Rapoport and Zink constructed rigid analytic period spaces $F^{wa}$ for Fontaine's filtered isocrystals, and period morphisms from PEL moduli spaces of $p$-divisible groups to some of these period spaces. They conjectured the existence of an étale bijective morphism $F^a \to F^{wa}$ of rigid analytic spaces and of a universal local system of $Q_p$-vector spaces on $F^a$. For Hodge-Tate weights $n-1$ and $n$ we construct in this article an intrinsic Berkovich open subspace $F^0$ of $F^{wa}$ and the universal local system on $F^0$. We conjecture that the rigid-analytic space associ…
Here, we utilize facts about the big Chern classes discovered by M. Kapranov and independently by M. V. Nori to prove certain results about the nonexistence of certain morphisms from Grassmannian to Grassmannian in characteristic 0. In particular, we show that the filtration F_. on CH^l(X) where F_rCH^l(X) is the linear span of ch_l(V) for all vector bundles V of rank <=r is nontrivial as a theory by showing that for a fixed l>=2, for infinitely many r, for a Grassmannian of r dimensional quotient spaces in an n dimensional vector space (n sufficiently large) over a field of charactersitic 0, …
An investigation of morphisms that coincide topologically is used to generalize to all characteristics and partly reprove Tamagawa's theorem on the Grothendieck conjecture in anabelian geometry for affine hyperbolic curves. The theorem now deals with π^{tame} of curves over a finitely generated field and its effect on the set of isomorphisms. Universal homeomorphisms are formally inverted.
