Effective Cartier Divisor Research Articles

Root stacks and periodic decompositions

For an effective Cartier divisor D on a scheme X we may form an nth\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${n}^{\ ext {th}}$$\\end{document} root stack. Its derived category is known to have a semiorthogonal decomposition with components given by D and X. We show that this decomposition is 2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2n$$\\end{document}-periodic. For n=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n=2$$\\end{document} this gives a purely triangulated proof of the existence of a known spherical functor, namely the pushforward along the embedding of D. For n>2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n > 2$$\\end{document} we find a higher spherical functor in the sense of recent work of Dyckerhoff et al. (N-spherical functors and categorification of Euler’s continuants. arXiv:2306.13350, 2023). We use a realization of the root stack construction as a variation of GIT, which may be of independent interest.

Cycle class maps for Chow groups of zero-cycles with modulus

For a smooth scheme X of pure dimension d over a field k and an effective Cartier divisor D⊂X whose support is a simple normal crossing divisor, we construct a cycle class mapcycX|D:CH0(X|D)→HNisd(X,KdM(OX,ID)) from the Chow group of zero-cycles with modulus to the cohomology of the relative Milnor K-sheaf.

Nevanlinna and algebraic hyperbolicity

Motivated by the notion of the algebraic hyperbolicity, we introduce the notion of Nevanlinna hyperbolicity for a pair [Formula: see text], where [Formula: see text] is a projective variety and [Formula: see text] is an effective Cartier divisor on [Formula: see text]. This notion links and unifies the Nevanlinna theory, the complex hyperbolicity (Brody and Kobayashi hyperbolicity), the big Picard-type extension theorem (more generally the Borel hyperbolicity). It also implies the algebraic hyperbolicity. The key is to use the Nevanlinna theory on parabolic Riemann surfaces recently developed by P[Formula: see text]un and Sibony [Value distribution theory for parabolic Riemann surfaces, preprint (2014), arXiv:1403.6596].

On Syntomic Complexes with Modulus for Semi-stable Reduction Cases

In this paper, we define a syntomic complex for modulus pair $(X,D)$, where $X$ is a regular semi-stable family and $D$ is an effective Cartier divisor on $X$ and we compute its cohomology sheaves. We construct a symbol map for this syntomic complex for modulus pair $(X,D)$ and investigate its cokernel by computing its cohomology sheaves. To compute the structure of the cohomology sheaves of syntomic complex with modulus, we define some filtrations on it.

Zero-cycles with modulus and relative K-theory

Let D be an effective Cartier divisor on a regular quasiprojective scheme X of dimension d≥1 over a field. For an integer n≥0, we construct a cycle class map from the higher Chow groups with modulus {CHn+d(X|mD,n)}m≥1 to the relative K-groups {Kn(X,mD)}m≥1 in the category of pro-abelian groups. We show that this induces a proisomorphism between the additive higher Chow groups of relative 0-cycles and the reduced algebraic K-groups of truncated polynomial rings over a regular semilocal ring which is essentially of finite type over a characteristic zero field.

Local topological obstruction for divisors

Given a smooth, projective variety $X$ and an effective divisor $D\,\subseteq\, X$, it is well-known that the (topological) obstruction to the deformation of the fundamental class of $D$ as a Hodge class, lies in $H^2(\mathcal{O}_X)$. In this article, we replace $H^2(\mathcal{O}_X)$ by $H^2_D(\mathcal{O}_X)$ and give an analogous topological obstruction theory. We compare the resulting local topological obstruction theory with the geometric obstruction theory (i.e., the obstruction to the deformation of $D$ as an effective Cartier divisor of a first order infinitesimal deformations of $X$). We apply this to study the jumping locus of families of linear systems and the Noether-Lefschetz locus. Finally, we give examples of first order deformations $X_t$ of $X$ for which the cohomology class $[D]$ deforms as a Hodge class but $D$ does not lift as an effective Cartier divisor of $X_t$.

Bounds on the torsion subgroups of Néron–Severi groups

Let X ↪ P r X \hookrightarrow \mathbb {P}^r be a smooth projective variety defined by homogeneous polynomials of degree ≤ d \leq d . We give an explicit upper bound on the order of the torsion subgroup ( NS ⁡ X ) t o r (\operatorname {NS} X)_{\mathrm {tor}} of the Néron–Severi group of X X . The bound is derived from an explicit upper bound on the number of irreducible components of the scheme C D i v n ⁡ X \operatorname {\mathbf {CDiv}}_n X parametrizing the effective Cartier divisors of degree n n on X X . We also give an upper bound on the number of generators of ( NS ⁡ X ) [ ℓ ∞ ] (\operatorname {NS} X)[\ell ^\infty ] uniform as ℓ ≠ c h a r k \ell \neq \mathrm {char} k varies.

The test function conjecture for local models of Weil-restricted groups

We prove the test function conjecture of Kottwitz and the first named author for local models of Shimura varieties with parahoric level structure attached to Weil-restricted groups, as defined by B. Levin. Our result covers the (modified) local models attached toallconnected reductive groups over$p$-adic local fields with$p\geqslant 5$. In addition, we give a self-contained study of relative affine Grassmannians and loop groups formed using general relative effective Cartier divisors in a relative curve over an arbitrary Noetherian affine scheme.

A birational Nevanlinna constant and its consequences

The purpose of this paper is to modify the notion of the Nevanlinna constant ${\rm Nev}(D)$ introduced by the first author for an effective Cartier divisor on a projective variety $X$. The modified notion is called the {\it birational Nevanlinna constant} and is denoted by ${\rm Nev}_{{\rm bir}}(D)$. The goal of ${\rm Nev}(D)$ and ${\rm Nev}_{{\rm bir}}(D)$ is to measure what is possible using the filtration method introduced by Faltings and W\"{u}stholz, and further developed by Corvaja and Zannier and, independently, by Evertse and Ferretti. By computing ${\rm Nev}_{{\rm bir}}(D)$ using subsequent work of Autissier, we establish a general result (see the General Theorem in Section 1), in both the arithmetic and complex cases, which extends the results of Evertse-Ferretti and of Ru to general divisors. The notion ${\rm Nev}_{{\rm bir}}(D)$ originally came from applications involving Weil functions, but it also can be defined in terms of local effectivity of Cartier divisors after lifting by a proper birational map.

Deformations of overconvergent isocrystals on the projective line

Let k be a perfect field of positive characteristic and Z an effective Cartier divisor in the projective line over k with complement U. In this note, we establish some results about the formal deformation theory of overconvergent isocrystals on U with fixed “local monodromy” along Z. En route, we show that a Hochschild cochain complex governs deformations of a module over an arbitrary associative algebra. We also relate this Hochschild cochain complex to a de Rham complex in order to understand the deformation theory of a differential module over a differential ring.

The Abel map for surface singularities I: generalities and examples

Let (X, o) be a complex normal surface singularity. We fix one of its good resolutions widetilde{X}rightarrow X, an effective cycle Z supported on the reduced exceptional curve, and any possible (first Chern) class l'in H^2(widetilde{X},{mathbb {Z}}). With these data we define the variety mathrm{ECa}^{l'}(Z) of those effective Cartier divisors D supported on Z, which determine a line bundles {{mathcal {O}}}_Z(D) with first Chern class l'. Furthermore, we consider the affine space mathrm{Pic}^{l'}(Z)subset H^1({{mathcal {O}}}_Z^*) of isomorphism classes of holomorphic line bundles with Chern class l' and the Abel map c^{l'}(Z):mathrm{ECa}^{l'}(Z)rightarrow mathrm{Pic}^{l'}(Z). The present manuscript develops the major properties of this map, and links them with the determination of the cohomology groups H^1(Z,{mathcal {L}}), where we might vary the analytic structure (X, o) (supported on a fixed topological type/resolution graph) and we also vary the possible line bundles {mathcal {L}}in mathrm{Pic}^{l'}(Z). The case of generic line bundles of mathrm{Pic}^{l'}(Z) and generic line bundles of the image of the Abel map will have priority roles. Rewriting the Abel map via Laufer duality based on integration of forms on divisors, we can make explicit the Abel map and its tangent map. The case of superisolated and weighted homogeneous singularities are exemplified with several details. The theory has similar goals (but rather different techniques) as the theory of Abel map or Brill–Noether theory of reduced smooth projective curves.

Normal hyperplane sections of normal schemes in mixed characteristic

The aim of this article is to prove that, under certain conditions, an affine flat normal scheme that is of finite type over a local Dedekind scheme in mixed characteristic admits infinitely many normal effective Cartier divisors. For the proof of this result, we prove the Bertini theorem for normal schemes of some type. We apply the main result to prove a result on the restriction map of divisor class groups of Grothendieck–Lefschetz type in mixed characteristic.

Torsion in the 0-cycle group with modulus

We show, for a smooth projective variety X over an algebraically closed field k with an effective Cartier divisor D, that the torsion subgroup CH0(X|D){l} can be described in terms of a relative etale cohomology for any prime l≠p= char(k). This extends a classical result of Bloch, on the torsion in the ordinary Chow group, to the modulus setting. We prove the Roitman torsion theorem (including p-torsion) for CH0(X|D) when D is reduced. We deduce applications to the problem of invariance of the prime-to-p torsion in CH0(X|D) under an infinitesimal extension of D.

Relative K0 and relative cycle class map

Let F:A→B be an exact functor between small exact categories. We study the zeroth homotopy group K0(F) of the homotopy fiber of the map K(A)→K(B) between K-theory spectra. Under the assumption that F is a cofinal and that B is split exact, we give an explicit description of K0(F) in terms of the triangulated functor Db(A)→Db(B) between the derived categories.We apply it to the pair (X,D) of a scheme X and an affine closed subscheme D of X, and get a description of the relative K0-group K0(X,D) in terms of perfect complexes; it is generated by pairs of two perfect complexes of X together with quasi-isomorphisms along D. This description makes it possible to assign a cycle class in K0(X,D) to a cycle on X not meeting D in an intuitive way. When X is a separated regular scheme of finite type over a field and D is an affine effective Cartier divisor on X, we prove that the cycle classes induce a surjective group homomorphism from the Chow group with modulus CH⁎(X|D) defined by Binda–Saito to a suitable subquotient of K0(X,D).

RELATIVE CYCLES WITH MODULI AND REGULATOR MAPS

Let$\overline{X}$be a separated scheme of finite type over a field$k$and$D$a non-reduced effective Cartier divisor on it. We attach to the pair$(\overline{X},D)$a cycle complex with modulus, those homotopy groups – called higher Chow groups with modulus – generalize additive higher Chow groups of Bloch–Esnault, Rülling, Park and Krishna–Levine, and that sheafified on$\overline{X}_{\text{Zar}}$gives a candidate definition for a relative motivic complex of the pair, that we compute in weight$1$. When$\overline{X}$is smooth over$k$and$D$is such that$D_{\text{red}}$is a normal crossing divisor, we construct a fundamental class in the cohomology of relative differentials for a cycle satisfying the modulus condition, refining El Zein’s explicit construction of the fundamental class of a cycle. This is used to define a natural regulator map from the relative motivic complex of$(\overline{X},D)$to the relative de Rham complex. When$\overline{X}$is defined over$\mathbb{C}$, the same method leads to the construction of a regulator map to a relative version of Deligne cohomology, generalizing Bloch’s regulator from higher Chow groups. Finally, when$\overline{X}$is moreover connected and proper over$\mathbb{C}$, we use relative Deligne cohomology to define relative intermediate Jacobians with modulus$J_{\overline{X}|D}^{r}$of the pair$(\overline{X},D)$. For$r=\dim \overline{X}$, we show that$J_{\overline{X}|D}^{r}$is the universal regular quotient of the Chow group of$0$-cycles with modulus.

Zero cycles with modulus and zero cycles on singular varieties

Given a smooth variety$X$and an effective Cartier divisor$D\subset X$, we show that the cohomological Chow group of 0-cycles on the double of$X$along$D$has a canonical decomposition in terms of the Chow group of 0-cycles$\text{CH}_{0}(X)$and the Chow group of 0-cycles with modulus$\text{CH}_{0}(X|D)$on$X$. When$X$is projective, we construct an Albanese variety with modulus and show that this is the universal regular quotient of$\text{CH}_{0}(X|D)$. As a consequence of the above decomposition, we prove the Roitman torsion theorem for the 0-cycles with modulus. We show that$\text{CH}_{0}(X|D)$is torsion-free and there is an injective cycle class map$\text{CH}_{0}(X|D){\hookrightarrow}K_{0}(X,D)$if$X$is affine. For a smooth affine surface$X$, this is strengthened to show that$K_{0}(X,D)$is an extension of$\text{CH}_{1}(X|D)$by$\text{CH}_{0}(X|D)$.

On a general Diophantine inequality

In [Ru15], the author introduced the notion of Nevanlinna constant (denoted by $Nev(D)$) for any effective Cartier divisor $D$ on a normal projective variety $X$, and established a defect relation for Zariski-dense holomorphic mappings $f: {\mathbb C}\rightarrow X$ in terms of $Nev(D)$. In this paper, we prove its counterpart result in Diophantine approximation, according to Vojta's correspondence (or Vojta's dictionary [Voj87]). The results obtained gave the quantitative extension of the earlier results of Corvaja-Zannier [CZ04a,CZ04b], Evertse-Ferretti [EF02, EF08], A. Levin [Lev09], P. Autissier [Aut1], and others.

Torsion 0-cycles with modulus on affine varieties

In this note, we show that given a smooth affine variety X over an algebraically closed field k and an effective (possibly non-reduced) Cartier divisor on it, the Chow group of zero cycles with modulus CH0(X|D) is torsion free, except possibly for p-torsion if the characteristic of k is p>0. This generalizes to the relative setting classical results of Rojtman (for X smooth) and of Levine (for X singular).

A Hermite–Minkowski type theorem of varieties over finite fields

We show the finiteness of étale coverings of a variety over a finite field with given degree whose ramification bounded along an effective Cartier divisor. The proof is an application of P. Delgine's theorem (H. Esnault and M. Kerz, 2012) [4] on a finiteness of l-adic sheaves with restricted ramification. By applying our result to a smooth curve over a finite field, we obtain a function field analogue of the classical Hermite–Minkowski theorem.

Computing Segre classes in arbitrary projective varieties

We give an algorithm for computing Segre classes of subschemes of arbitrary projective varieties by computing degrees of a sequence of linear projections. Based on the fact that Segre classes of projective varieties commute with intersections by general effective Cartier divisors, we can compile a system of linear equations which determine the coefficients for the Segre class pushed forward to projective space. The algorithm presented here comes after several others which solve the problem in special cases, where the ambient variety is for instance projective space; to our knowledge, this is the first algorithm to be able to compute Segre classes in projective varieties with arbitrary singularities.