Working over a ground field of characteristic zero, this paper studies the quotient morphism $π:X\to Y$ for an affine $\mathbb{G}_a$-variety $X$ with affine quotient $Y$. It is shown that the degree modules associated to the $\mathbb{G}_a$-action give a uniquely determined sequence of dominant $\mathbb{G}_a$-equivariant morphisms, $X=X_r\to X_{r-1}\to\cdots\to X_1\to X_0=Y$, where $X_i$ is an affine $\mathbb{G}_a$-variety and $X_{i+1}\to X_i$ is birational for each $i\ge 1$. This is the canonical factorization of $π$. We give an algorithm for finding the degree modules associated to the given $\mathbb{G}_a$-action, and this yields the canonical factorization of the quotient morphism. The algorithm is applied to compute the canonical factorization for several examples, including the homogeneous $(2,5)$-action on $\mathbb{A}^3$. By a fundamental result of Kaliman and Zaidenberg, any birational morphism of affine varieties is an affine modification, and each mapping in these examples is presented as a $\mathbb{G}_a$-equivariant affine modification.
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.
We study mixed surfaces, the minimal resolution S of the singularities of a quotient (C x C)/G of the "square" of a curve by a finite group G of automorphisms that contains elements not preserving the factors. We study them through the "further quotients" by (C x C)/G' where G' contains G. As a first application we prove that if the irregularity is at least 3, then S is also minimal. The result is sharp. The main result is a complete description of the Albanese morphism of S through a determined further quotient (C x C)/G' that is an étale cover of the symmetric square of a curve. In particula…
This is the second in a sequence of papers in which we construct a quantum version of the Kirwan map from the equivariant quantum cohomology of a smooth polarized complex projective variety with the action of a connected complex reductive group to the orbifold quantum cohomology of its geometric invariant theory quotient, and prove that it intertwines the genus zero gauged Gromov-Witten potential with the genus zero Gromov-Witten graph potential. In this part we construct virtual fundamental classes on the moduli spaces used in the construction of the quantum Kirwan map and the gauged Gromov-W…
The purpose of this paper is to develop an efficient computational model for Abelian categories of coherent sheaves over certain classes of varieties. These categories are naturally described as Serre quotient categories. Hence, our approach relies on describing general Serre quotient categories in a constructive way which leads to an efficient computer implementation.
This is the third in a sequence of papers in which we construct a quantum version of the Kirwan map from the equivariant quantum cohomology of a smooth polarized complex projective variety with the action of a connected complex reductive group to the orbifold quantum cohomology of its geometric invariant theory quotient, and prove that it intertwines the genus zero gauged Gromov-Witten potential with the genus zero Gromov-Witten graph potential. We also give a formula for a solution to the quantum differential equation in terms of a localized gauged potential. These results overlap with those …
We give some necessary and sufficient conditions for the existence of Néron models of jacobians of semistable morphisms of arbitrary relative dimension over base schemes of arbitrary dimension. To do this, we introduce a notion of alignment for semistable morphisms over any regular base scheme, and show that the jacobian of an aligned projective semistable morphism admits a separated model with the Néron mapping property. When the Picard scheme is smooth over the base scheme along its unit section we show that the converse holds.
We prove that, for certain extensions of valued fields which admit a sensible theory of ramification groups, there exist canonical towers that correspond to the break-points of their Herbrand function. In particular, each of the intermediate field extensions in the tower has a Herbrand function with only one break-point and there is at most one extension with trivial Herbrand function. We apply the result to the setting of finite morphisms of Berkovich curves where we prove the existence of canonical local and global factorization of such morphisms according to their metric properties. Finally…
We transfer several elementary geometric properties of rigid-analytic spaces to the world of adic spaces, more precisely to the category of adic spaces which are locally of (weakly) finite type over a non-archimedean field. This includes normality, irreducibility (in particular irreducible components) and a Stein factorization theorem. Most notably we show that finite morphisms of adic spaces are open under mild assumptions on the base and target space.
We prove some fundamental results like localization, excision, Nisnevich descent and the Mayer-Vietoris property for equivariant regular blow-up for the equivariant K-theory of schemes with an affine group scheme action. We also show that the equivariant K-theory with finite coefficients is invariant under equivariant vector bundle morphisms. We show that the equivariant homotopy K-theory is invariant under equivariant vector bundle morphisms and satisfies all the above properties, including nil-invariance.
We study GIT quotients parametrizing n-pointed conics that generalize the GIT quotients $(\mathbb{P}^1)^n//SL2$. Our main result is that $\overline{M}_{0,n}$ admits a morphism to each such GIT quotient, analogous to the well-known result of Kapranov for the simpler $(\mathbb{P}^1)^n$ quotients. Moreover, these morphisms factor through Hassett's moduli spaces of weighted pointed rational curves, where the weight data comes from the GIT linearization data.
This is the first in a sequence of papers in which we construct a quantum version of the Kirwan map from the equivariant quantum cohomology $QH_G(X)$ of a smooth complex projective variety X with the action of a connected complex reductive group $G$ to the orbifold quantum cohomology $QH(X//G)$ of its geometric invariant theory quotient $X//G$, and prove that it intertwines the genus zero gauged Gromov-Witten potential of X with the genus zero Gromov-Witten graph potential of $X//G$.
Given a variety defined over a field of characteristic zero and an algebraically integrable foliation of corank less than or equal to two, we show the existence of a categorical quotient, defined on the non-empty open set of stable points, through which every invariant morphism factors uniquely.
Motivated by the study of the structure of algebraic actions the additive group on affine threefolds X, we consider a special class of such varieties whose algebraic quotient morphisms X $\rightarrow$ X//Ga restrict to principal homogeneous bundles over the complement of a smooth point of the quotient. We establish basic general properties of these varieties and construct families of examples illustrating their rich geometry. In particular, we give a complete classification of a natural subclass consisting of threefolds X endowed with proper Ga-actions, whose algebraic quotient morphisms $π$ :…
Let X be a smooth projective variety with the action of the n dimensional torus. The article describes the moduli space of torus equivariant morphisms from stable toric varieties into X as the inverse limit of the GIT quotients of X and their flips when these spaces are enhanced with a naturally associated Deligne-Mumford stack structure. This description is used for constructing a class in the Chow group of the moduli space of dimension dim(X)-n which is invariant to equivariant deformations of X.
Affine varieties of dimension greater than two can be explored their structures with the help of fibrations by the affine line or plane and quotient morphisms by $\mathbb{G}_a$-actions. We consider $\mathbb{G}_a$-actions on affine threefolds and discuss the structure and the singularities of the quotient surface as well as the singular fibers of the quotient morphism. Relative $\mathbb{G}_a$ and $\mathbb{G}_m$-actions on $\mathbb{A}^4$ or on an affine 4-fold having $\mathbb{A}^3$-fibrations over $\mathbb{A}^1$ are discussed when the actions leave one variable invariant or the fibration invaria…
The aim of this article is to make a first step towards the classification of complex normal affine $\mathbb G_a$-threefolds $X$. We consider the case where the restriction of the quotient morphism $π\colon X\to S$ to $π^{-1}(S_*)$, where $S_*$ denotes the complement of some regular closed point in $S$, is a principal $\mathbb G_a$-bundle. The variety $\mathrm{SL}_2$ will be of special interest and a source of many examples. It has a natural right $\mathbb G_a$-action such that the quotient morphism $\mathrm{SL}_2\to\mathbb A^2$ restricts to a principal $\mathbb G_a$-bundle over the punctured …
We provide a direct proof, valid in arbitrary characteristic, of the result originally proven by Kapranov over $\mathbb{C}$, that the Hilbert and Chow quotients $(\mathbb{P}^1)^n//PGL2$ are isomorphic to $\overline{M}_{0,n}$. In both cases this is done by explicitly constructing the universal family and then showing that the induced morphism is an isomorphism onto its image. The proofs of these results in many ways reduce to the case $n = 4$; in an appendix we outline a formalism of this phenomenon relating to certain operads.
We study relatively affine actions of a diagonalizable group $G$ on locally noetherian schemes. In particular, we generalize Luna's fundamental lemma when applied to a diagonalizable group: we obtain criteria for a $G$-equivariant morphism $f: X'\to X$ to be $strongly\ equivariant$, namely the base change of the morphism $f/\!/G$ of quotient schemes, and establish descent criteria for $f/\!/G$ to be an open embedding, étale, smooth, regular, syntomic, or lci.
We study the Borel moment map $μ_B:T^*(\mathfrak{b}\times \mathbb{C}^n)\rightarrow \mathfrak{b}^*$, given by $(r,s,i,j)\mapsto [r,s]+ij$, and describe our algorithm to construct the geometric invariant theory (GIT) quotients $μ_B^{-1}(0)/\!\!/_{\det}B$ and $μ_B^{-1}(0)/\!\!/_{\det^{-1}}B$, and the affine quotient $μ_B^{-1}(0)/\!\!/B$. We also provide an insight of the singular locus of $2^n$ irreducible components of $μ_B$. Finally, analogous to the Hilbert--Chow morphism, we discuss that the GIT quotient for the Borel setting is a resolution of singularities.
