In this paper we study relative Riemann-Zariski spaces attached to a morphism of schemes and generalizing the classical Riemann-Zariski space of a field. We prove that similarly to the classical RZ spaces, the relative ones can be described either as projective limits of schemes in the category of locally ringed spaces or as certain spaces of valuations. We apply these spaces to prove the following two new results: a strong version of stable modification theorem for relative curves; a decomposition theorem which asserts that any separated morphism between quasi-compact and quasi-separated schemes factors as a composition of an affine morphism and a proper morphism. (In particular, we obtain a new proof of Nagata's compactification theorem.)
We show that any pseudo-effective divisor on a normal surface decomposes uniquely into its "integral positive" part and "integral negative" part, which is an integral analog of Zariski decompositions. By using this decomposition, we give three applications: a vanishing theorem of divisors on surfaces (a generalization of Kawamata-Viehweg and Miyaoka vanishing theorems), Reider-type theorems of adjoint linear systems on surfaces (including a log version and a relative version of the original one) and extension theorems of morphisms defined on curves on surfaces (generalizations of Serrano and P…
For $n\ge5$, it is well known that the moduli space $\mathfrak{M_{0,\:n}}$ of unordered $n$ points on the Riemann sphere is a quotient space of the Zariski open set $K_n$ of $\mathbb C^{n-3}$ by an $S_n$ action. The stabilizers of this $S_n$ action at certain points of this Zariski open set $K_n$ correspond to the groups fixing the sets of $n$ points on the Riemann sphere. Let $α$ be a subset of $n$ distinct points on the Riemann sphere. We call the group of all linear fractional transformations leaving $α$ invariant the stabilizer of $α$, which is finite by observation. For each non-trivial f…
We give a new proof of Raynaud-Gruson's theorem regarding flattening by blow-ups. The proof is direct, by working directly on the inverse limit of admissible blow-ups, which is a valuative space similar to the classical Zariski-Riemann space. These valuative spaces are defined in a broader context and always have an analogous flattening property: this yields a generalization of Raynaud-Gruson's theorem.
We prove that the base space of a log smooth family of log canonical pairs of log general type is of log general type as well as algebraically degenerate, when the family admits a relative good minimal model over a Zariski open subset of the base and the relative log canonical model is of maximal variation.
Let $S$ be a finite subset of a compact connected Riemann surface $X$ of genus $g \geq 2$. Let $\cat{M}_{lc}(n,d)$ denote the moduli space of pairs $(E,D)$, where $E$ is a holomorphic vector bundle over $X$ and $D$ is a logarithmic connection on $E$ singular over $S$, with fixed residues in the centre of $\mathfrak{gl}(n,\C)$, where $n$ and $d$ are mutually corpime. Let $L$ denote a fixed line bundle with a logarithmic connection $D_L$ singular over $S$. Let $\cat{M}'_{lc}(n,d)$ and $\cat{M}_{lc}(n,L)$ be the moduli spaces parametrising all pairs $(E,D)$ such that underlying vector bundle $E$ …
Let $X$ be a compact Riemann surface of genus $g \geq 3$ and $S$ a finite subset of $X$. Let $ξ$ be fixed a holomorphic line bundle over $X$ of degree $d$. Let $\mathcal{M}_{pc}(r, d, α)$ (respectively, $\mathcal{M}_{pc}(r, α, ξ)$ ) denote the moduli space of parabolic connections of rank $r$, degree $d$ and full flag rational generic weight system $α$, (respectively, with the fixed determinant $ξ$) singular over the parabolic points $S \subset X$. Let $\mathcal{M}'_{pc}(r, d, α)$ (respectively, $\mathcal{M}'_{pc}(r, α, ξ)$) be the Zariski dense open subset of $\mathcal{M}_{pc}(r, d, α)$ (resp…
We introduce the notion of strong regular holonomic ${\mathcal{D}}_{{X\times S}/S}$-module and we prove that the functor ${\mathrm{RH}}^S$ introduced by T. Monteiro Fernandes and C. Sabbah in [14] takes image in ${\mathsf{D}}^{\mathrm{b}}_{\mathrm{srhol}}({\mathcal{D}}_{{X\times S}/S})$ (complexes of ${\mathcal{D}}_{{X\times S}/S}$-module whose cohomologies are strongly regular). We prove that for $\dim X=\dim S=1$ the functor solution functor ${}^\mathrm{p}{\mathrm{Sol}}$ restricted to ${\mathsf{D}}^{\mathrm{b}}_{\mathrm{srhol}}({\mathcal{D}}_{{X\times S}/S})$ is an equivalence of categories …
In this paper, we prove the functorial Riemann-Roch theorem in positive characteristic for a smooth and projective morphism with any relative dimension. In the case of relative dimension $1$, we have given an analogue with Deligne's functorial Riemann Roch theorem in previous author's paper. For any relative dimension, our result can deduce an analogue to the Knudsen-Mumford extension. The present result is a generalization, which mainly originated from the extended Deligne pairing by S. Zhang and the Adams Riemann Roch theorem in positive characteristic by R. Pink and D. Rössler.
For any nonconstant f,g in C(x) such that the numerator H(x,y) of f(x)-g(y) is irreducible, we compute the genus of the normalization of the curve H(x,y)=0. We also prove an analogous formula in arbitrary characteristic when f and g have no common wildly ramified branch points, and generalize to (possibly reducible) fiber products of nonconstant morphisms of curves f:A-->D and g:B-->D.
On the product of a complex manifold $X$ by a complex curve $S$ considered as a parameter space, we show a Riemann-Hilbert correspondence between regular holonomic relative $\mathcal D$-modules (resp. complexes) on the one hand and relative perverse complexes (resp. $S$-$\mathbb{C}$-constructible complexes) on the other hand.
Relative $\mathbb{A}^1$-Contractibility of Smooth Schemes
CNRS, Université de Poitiers, LMA（法国国家科学研究中心，普瓦捷大学，数学实验室）；Université Bourgogne Europe, CNRS, IMB UMR 5584（勃艮第欧洲大学，法国国家科学研究中心，IMB UMR 5584）；Department of Mathematics “F. Enriques”, University of Milan（米兰大学，“F.恩里克斯”数学系）
