Root Stack 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.

Building data for stacky covers

We define stacky building data for stacky covers in the spirit of Pardini and give an equivalence of (2,1)-categories between the category of stacky covers and the category of stacky building data. We show that every stacky cover is a flat root stack in the sense of Olsson and Borne–Vistoli and give an intrinsic description of it as a root stack using stacky building data. When the base scheme S is defined over a field, we give a criterion for when a birational building datum comes from a tamely ramified cover for a finite abelian group scheme, generalizing a result of Biswas–Borne.

On parahoric Hitchin systems over curves

In this paper, we talk about parahoric Hitchin systems over smooth projective curves with the structure group a semisimple simply connected group. We prove the equivalence of parahoric Hitchin systems over the curve with Hitchin systems over a corresponding root stack with a finite cyclic group action determined (up to conjugation) by the parahoric data. And we also show the compatibility of the equivalence with Hitchin maps. We work over an algebraically closed field with a mild assumption of the characteristic.

VELOCITY ANALYSIS USING N-TH ROOT STACK

The estimation of seicmic wave velocity in the subsurface is one of the most important process in seismic data processing. The result is very crucial for the conversion af seismic data fromtime section into depth section. The N-th Root Stack which has not been widely used in the industry in fact a powerful method for estimating seismic wave velocity from seicmic record. The method attenuates the background/random noise effectiviely which facilitates tre fitting of the velocity spectrum can be done anccurately. In addition, the noise is also effectivdy suppressed during the CDP stackins, yielding the clear inrim i doking amplitude Synthetic records have been used to demonstrate the effectiveness of the N-th Root Stack Comparison with the Delay and Sum method also demonstrate the superiority of the N-th Root Stack.

Velocity Analysis Using N-Th Root Stack

The estimation of seicmic wave velocity in the subsurface is one of the most important process in seismic data processing. The result is very crucial for the conversion af seismic data fromtime section into depth section. The N-th Root Stack which has not been widely used in the industry in fact a powerful method for estimating seismic wave velocity from seicmic record. The method attenuates the background/random noise effectiviely which facilitates tre fitting of the velocity spectrum can be done anccurately. In addition, the noise is also effectivdy suppressed during the CDP stackins, yielding the clear inrim i doking amplitude Synthetic records have been used to demonstrate the effectiveness of the N-th Root Stack Comparison with the Delay and Sum method also demonstrate the superiority of the N-th Root Stack.

Cohomological invariants of root stacks and admissible double coverings

AbstractWe give a formula for the cohomological invariants of a root stack, which we apply to compute the cohomological invariants and the Brauer group of the compactification of the stacks of hyperelliptic curves given by admissible double coverings.

Artin–Schreier root stacks

We classify stacky curves in characteristic p>0 with cyclic stabilizers of order p using higher ramification data. This approach replaces the local root stack structure of a tame stacky curve, similar to the local structure of a complex orbifold curve, with a more sensitive structure called an Artin–Schreier root stack, allowing us to incorporate this ramification data directly into the stack. As an application, we compute dimensions of Riemann–Roch spaces for some examples of stacky curves in positive characteristic and suggest a program for computing spaces of modular forms in this setting.

The ring of modular forms for the even unimodular lattice of signature (2,10)

We show that the ring of modular forms with characters for the even unimodular lattice of signature (2,10) is generated by forms of weights 4, 10, 12, 16, 18, 22, 24, 28, 30, 36, 42, and 252 with one relation of weight 504. The proof is based on the comparison of the orbifold quotient of the symmetric domain with the root stack of the coarse moduli space.

Gromov–Witten invariants of root stacks with mid-ages and the loop axiom

We study orbifold Gromov–Witten invariants of the r-th root stack XD,r with a pair of mid-ages when r is sufficiently large. We prove that genus g invariants with a pair of mid-ages ka/r and 1−ka/r are polynomials in ka and the kai-coefficients are polynomials in r with degree bounded by 2g. Moreover, genus zero invariants with a pair of mid-ages are independent of the choice of mid-ages. As an application, we obtain an identity for relative Gromov–Witten theory which can be viewed as a modified version of the usual loop axiom.

The Tanaka–Thomas’s Vafa–Witten invariants via surface Deligne–Mumford stacks

We provide a definition of Vafa-Witten invariants for projective surface Deligne-Mumford stacks, generalizing the construction of Tanaka-Thomas on the Vafa-Witten invariants for projective surfaces inspired by the S-duality conjecture. We give calculations for a root stack over a general type quintic surface, and quintic surfaces with ADE singularities. The relationship between the Vafa-Witten invariants of quintic surfaces with ADE singularities and the Vafa-Witten invariants of their crepant resolutions is also discussed.

Higher genus relative and orbifold Gromov–Witten invariants

Given a smooth projective variety $X$ and a smooth divisor $D\subset X$. We study relative Gromov-Witten invariants of $(X,D)$ and the corresponding orbifold Gromov-Witten invariants of the $r$-th root stack $X_{D,r}$. For sufficiently large $r$, we prove that orbifold Gromov-Witten invariants of $X_{D,r}$ are polynomials in $r$. Moreover, higher genus relative Gromov-Witten invariants of $(X,D)$ are exactly the constant terms of the corresponding higher genus orbifold Gromov-Witten invariants of $X_{D,r}$. We also provide a new proof for the equality between genus zero relative and orbifold Gromov-Witten invariants, originally proved by Abramovich-Cadman-Wise \cite{ACW}. When $r$ is sufficiently large and $X=C$ is a curve, we prove that stationary relative invariants of $C$ are equal to the stationary orbifold invariants in all genera.

The FFRT property of two-dimensional normal graded rings and orbifold curves

We study the finite F-representation type (abbr. FFRT) property of a two-dimensional normal graded ring R in characteristic p>0, using notions from the theory of algebraic stacks. Given a graded ring R, we consider an orbifold curve C, which is a root stack over the smooth curve C=ProjR, such that R is the section ring associated with a line bundle L on C. The FFRT property of R is then rephrased with respect to the Frobenius push-forwards F⁎e(Li) on the orbifold curve C. As a result, we see that if the singularity of R is not log terminal, then R has FFRT only in exceptional cases where the characteristic p divides a weight of C.

Gushel–Mukai varieties: Moduli

We describe the moduli stack of Gushel–Mukai varieties as a global quotient stack and its coarse moduli space as the corresponding GIT quotient. The construction is based on a comprehensive study of the relation between this stack and the stack of so-called Lagrangian data defined in our previous works; roughly speaking, we show that the former is a generalized root stack of the latter. As an application, we define the period map for Gushel–Mukai varieties and construct some complete nonisotrivial families of smooth Gushel–Mukai varieties. In an appendix, we describe a generalization of the root stack construction used in our approach to the moduli stack.

The circle quantum group and the infinite root stack of a curve

In the present paper, we give a definition of the quantum group $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1))$ of the circle $S^1\colon =\mathbb{R}/\mathbb{Z}$, and its fundamental representation. Such a definition is motivated by a realization of a quantum group $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1_\mathbb{Q}))$ associated to the rational circle $S^1_\mathbb{Q}\colon= \mathbb{Q}/\mathbb{Z}$ as a direct limit of $\mathbf{U}_\upsilon(\widehat{\mathfrak{sl}}(n))$'s, where the order is given by divisibility of positive integers. The quantum group $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1_\mathbb{Q}))$ arises as a subalgebra of the Hall algebra of coherent sheaves on the infinite root stack $X_\infty$ over a fixed smooth projective curve $X$ defined over a finite field. Via this Hall algebra approach, we are able to realize geometrically the fundamental and the tensor representations, and a family of symmetric tensor representations, depending on the genus $g_X$, of $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1_\mathbb{Q}))$. Moreover, we show that $\mathbf{U}_\upsilon(\widehat{\mathfrak{sl}}(+\infty))$ and $\mathbf{U}_\upsilon(\widehat{\mathfrak{sl}}(\infty))$ are subalgebras of $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1_\mathbb{Q}))$. As proved by T. Kuwagaki in the appendix, the quantum group $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1))$ naturally arises as well in the mirror dual picture, as a Hall algebra of constructible sheaves on the circle $S^1$.

Mirror theorems for root stacks and relative pairs

Given a smooth projective variety $X$ with a smooth nef divisor $D$ and a positive integer $r$, we construct an $I$-function, an explicit slice of Givental's Lagrangian cone, for Gromov--Witten theory of the root stack $X_{D,r}$. As an application, we also obtain an $I$-function for relative Gromov--Witten theory following the relation between relative and orbifold Gromov--Witten invariants.

G-theory of root stacks and equivariant K-theory

Using the description of the category of quasi-coherent sheaves on a root stack given in the paper of N. Borne and A. Vistoli, we study the G-theory of root stacks via localisation methods. We apply our results to the study of equivariant K-theory of algebraic varieties under certain conditions.

On a logarithmic version of the derived McKay correspondence

We globalize the derived version of the McKay correspondence of Bridgeland, King and Reid, proven by Kawamata in the case of abelian quotient singularities, to certain logarithmic algebraic stacks with locally free log structure. The two sides of the correspondence are given respectively by the infinite root stack and by a certain version of the valuativization (the projective limit of every possible logarithmic blow-up). Our results imply, in particular, that in good cases the category of coherent parabolic sheaves with rational weights is invariant under logarithmic blow-up, up to Morita equivalence.

Geometric Brauer residue via root stacks

We reinterpret the residue map for the Brauer group of a smooth variety using a root stack construction and Weil restriction for algebraic stacks, and apply the result to find a geometric representative of for the residue of a Brauer class associated to a conic bundle, or more generally a Brauer-Severi bundle.

Infinite root stacks and quasi-coherent sheaves on logarithmic schemes

We define and study infinite root stacks of fine and saturated logarithmic schemes, a limit version of the root stacks introduced by Niels Borne and the second author. We show in particular that the infinite root stack determines the logarithmic structure, and recovers the Kummer-flat topos of the logarithmic scheme. We also extend the correspondence between parabolic sheaves and quasi-coherent sheaves on root stacks to this new setting.

Relative and orbifold Gromov�Witten invariants

We prove that genus zero Gromov--Witten invariants of a smooth scheme relative to a smooth divisor coincide with genus zero orbifold Gromov--Witten invariants of an appropriate root stack construction along the divisor.