The Fulton-MacPherson compactification is not a Mori dream space

The Chow ring of relative Fulton-MacPherson space

Suppose that X is a nonsingular variety and D is a nonsingular proper subvariety. Configuration spaces of distinct and non-distinct n points in X away from D were constructed by the author and B. Kim in arXiv:0806.3819, by using the method of wonderful compactification. In this paper, we give an explicit presentation of Chow motives and Chow rings of these configuration spaces.

On the biregular geometry of the Fulton-MacPherson compactification

Let $X[n]$ be the Fulton-MacPherson compactification of the configuration space of $n$ ordered points on a smooth projective variety $X$. We prove that if either $n\neq 2$ or $\dim(X)\geq 2$, then the connected component of the identity of $Aut(X[n])$ is isomorphic to the connected component of the identity of $Aut(X)$. When $X = C$ is a curve of genus $g(C)\neq 1$ we classify the dominant morphisms $C[n]\rightarrow C[r]$, and thanks to this we manage to compute the whole automorphism group of $C[n]$, namely $Aut(C[n])\cong S_n\times Aut(C)$ for any $n\neq 2$, while $Aut(C[2])\cong S_2\ltimes …

A Geometric Invariant Theory Compactification of M_{g,n} Via the Fulton-MacPherson Configuration Space

A compactification over $\overline{M}_g$ of $M_{g,n}$ is obtained by considering the relative Fulton-MacPherson configuration space of the universal curve. The resulting compactification differs from the Deligne-Mumford space $\overline{M}_{g,n}$. In case $n=2$, the compactification constructed here and the Deligne-Mumford compactification are essentially the distinct minimal resolutions of the fiber product over $\overline{M}_g$ of the universal curve with itself.

Balanced normal cones and Fulton-MacPherson's intersection theory

Let X be a subscheme of a reduced scheme Y. Then Y has a flat "degeneration to the normal cone" C_X Y of X, and this degeneration plays a key step in Fulton and MacPherson's "basic construction" in intersection theory. The intersection product has a canonical refinement as a sum over the components of C_X Y, for X and Y depending on the given intersection problem. The cone C_X Y is usually not reduced, which leads to the appearance of multiplicities in intersection formulae. We describe a variant of this degeneration, due essentially to Samuel, Rees, and Nagata, in which Y flatly degenerates t…

On CSM classes via Chern-Fulton classes of f-schemes

The Chern-Fulton class is a generalization of Chern class to the realm of arbitrary embeddable schemes. While Chern-Fulton classes are sensitive to non-reduced scheme structure, they are not sensitive to possible singularities of the underlying support, thus at first glance are not interesting from a singularity theory viewpoint. However, we introduce a class of formal objects which we think of as `fractional schemes', or f-schemes for short, and then show that when one broadens the domain of Chern-Fulton classes to f-schemes one may indeed recover singularity invariants of varieties and schem…

Polydiagonal compactification of configuration spaces

A smooth compactification X<n> of the configuration space of n distinct labeled points in a smooth algebraic variety X is constructed by a natural sequence of blowups, with the full symmetry of the permutation group S_n manifest at each stage of the construction. The strata of the normal crossing divisor at infinity are labeled by trees with levels and their structure is studied. This is the maximal wonderful compactification in the sense of DeConcini-Procesi, and it has a strata-compatible surjection onto the Fulton-MacPherson compactification. The degenerate configurations added in the compa…

The chow cohomology of affine toric varieties

We study the Fulton-Macpherson Chow cohomology of affine toric varieties. In particular, we prove that the Chow cohomology vanishes in positive degree. We prove an analogous result for the operational $K$-theory defined by Anderson and Payne.

Weighted compactifications of configuration spaces and relative stable degenerations

We study a compactification of the configuration space of n distinct labeled points on an arbitrary nonsingular variety. Our construction provides a generalization of the original Fulton-MacPherson compactification that is parallel to the generalization of the moduli space of n-pointed stable curves carried out by Hassett.

Residual Intersections and Some Applications

We give a new residual intersection decomposition for the refined intersection products of Fulton-MacPherson. Our formula refines the celebrated residual intersection formula of Fulton, Kleiman, Laksov, and MacPherson. The new decomposition is more likely to be compatible with the canonical decomposition of the intersection products and each term in the decomposition thus has simple geometric meaning. Our study is motivated by its applications to some geometric problems. In particular, we use the decomposition to find the distribution of limiting linear subspaces in degenerations of hypersurfa…

Motivic Milnor classes

The Milnor class is a generalization of the Milnor number, defined as the difference (up to sign) of Chern--Schwartz--MacPherson's class and Fulton--Johnson's canonical Chern class of a local complete intersection variety in a smooth variety. In this paper we introduce a "motivic" Grothendieck group $K^{\mathcal Prop}_{\ell.c.i}(\mathcal V/X \to S)$ and natural transformations from this Grothendieck group to the homology theory. We capture the Milnor class, more generally Hirzebruch--Milnor class, as a special value of a distinguished element under these natural transformations. We also show a…

Bivariant Chern classes and Grothendieck transformations

The existence of bivariant Chern classes was conjectured by W.Fulton and R.MacPherson and proved by J.P.Brasselet for cellular morphisms of analytic varieties. In this paper we show that restricted to morphisms whose target varieties are possible singular but (rational) homology manifolds, the bivariant Chern classes (with rational coefficients) are uniquely determined. In the final sections we construct a unique bivariant Chern class satisfying a suitable normalization condition. In fact, it will be a special case of a general construction of unique Grothendieck transformations, which in a se…

曲面希尔伯特概形的$\boldsymbol{\text{CohFT}}$的普适性

Universality of $\mathsf{CohFT}$ of $\mathsf{Hilb}_n(S)$

该研究确定曲面希尔伯特概形的上同调场论在极端曲线类中由曲面特征数与仿射平面希尔伯特概形理论普适决定，并据此算出所有曲面希尔伯特概形的亏格1极端Gromov–Witten不变量。

Wonderful compactifications of the moduli space of points in affine and projective space

We introduce and study smooth compactifications of the moduli space of n labeled points with weights in projective space, which have normal crossings boundary and are defined as GIT quotients of the weighted Fulton-MacPherson compactification. We show that the GIT quotient of a wonderful compactification is also a wonderful compactification under certain hypotheses. We also study a weighted version of the configuration spaces parametrizing n points in affine space up to translation and homothety. In dimension one, the above compactifications are isomorphic to Hassett's moduli space of rational…

Ordinarity of configuration spaces and of wonderful compactifications

We prove the following: (1) if $X$ is ordinary, the Fulton-MacPherson configuration space $X[n]$ is ordinary for all $n$; (2) the moduli of stable $n$-pointed curves of genus zero is ordinary. (3) More generally we show that a wonderful compactification $X_\sg$ is ordinary if and only if $(X,\sg)$ is an ordinary building set. This implies the ordinarity of many other well-known configuration spaces (under suitable assumptions).

Integral points and effective cones of moduli spaces of stable maps

Consider the Fulton-MacPherson configuration space of $n$ points on $¶^1$, which is isomorphic to a certain moduli space of stable maps to $¶^1$. We compute the cone of effective ${\mathfrak S}_n$-invariant divisors on this space. This yields a geometric interpretation of known asymptotic formulas for the number of integral points of bounded height on compactifications of $\SL_2$ in the space of binary forms of degree $n\ge 3$.

Mixed Hodge structures of configuration spaces

The symmetric group S_n acts freely on the configuration space of n distinct points in a quasi-projective variety. In this paper, we study the induced action of the symmetric group S_n on the de Rham cohomology of this space, using mixed Hodge theory, combined with methods from the theory of symmetric functions. (We prove a motivic version of this as well.) As an application of our results, we calculate the S_n-equivariant Hodge polynomial of the Fulton-MacPherson compactification X[n] of the configuration space.

Multiplicative models for configuration spaces of algebraic varieties

W. Fulton and R. MacPherson described a Sullivan dg-algebra model for the space of n-configurations of labeled points in a smooth compact complex algebraic variety X. I. Kriz then gave a simpler model that depends only on the cohomology ring of X. We construct an even simpler and substantially smaller model for this configuration space. Using this smaller model, we also define another, new dg-algebra model, for the space of n-configurations in the variety X with a point removed. Following an idea of V.G. Drinfel'd, we endow the collection of these new "punctured" models with a simplicial bigra…

Modular interpretation of a non-reductive Chow quotient

The space of n distinct points and a disjoint parameterized hyperplane in projective d-space up to projectivity---equivalently, configurations of n distinct points in affine d-space up to translation and homothety---has a beautiful compactification introduced by Chen-Gibney-Krashen. This variety, constructed inductively using the apparatus of Fulton-MacPherson configuration spaces, is a parameter space of certain pointed rational varieties whose dual intersection complex is a rooted tree. This generalizes $\overline{M}_{0,n}$ and shares many properties with it. In this paper, we prove that the…

Cech cover of the complement of the discriminant variety. Part II: Deformations of Gauss-skizze

The configuration space of $n$ marked points on the complex plane is considered. We investigate a decomposition of this space by so-called Gauss-skizze i.e. a class of graphs being forests, introduced by Gauss. It is proved that this decomposition is a semi-algebraic topological stratification. It also forms a cell decomposition of the configuration space of $n$ marked points. Moreover, we prove that classical tools from deformation theory, ruled by a Maurer--Cartan equation, can be used only locally for Gauss-skizze. We prove that the deformation of the Gauss-skizze is governed by a Hamilton-…

Limits of Voronoi Diagrams

In this thesis we study sets of points in the plane and their Voronoi diagrams, in particular when the points coincide. We bring together two ways of studying point sets that have received a lot of attention in recent years: Voronoi diagrams and compactifications of configuration spaces. We study moving and colliding points and this enables us to introduce `limit Voronoi diagrams'. We define several compactifications by considering geometric properties of pairs and triples of points. In this way we are able to define a smooth, real version of the Fulton-MacPherson compactification. We show how…

The Space of Triangles, Vanishing Theorems, and Combinatorics

We consider compactifications of the space of triples of distinct points in projective $n$-space. One such space is a singular variety of configurations of points and lines; another is the smooth compactification of Fulton and MacPherson; and a third is the triangle space of Schubert and Semple. We compute the sections of line bundles on these spaces, and show that they are equal as GL(n) representations to the generalized Schur modules associated to ``bad'' generalized Young diagrams with three rows (Borel-Weil theorem). On the one hand, this yields Weyl-type character and dimension formulas …

Polymatroids 与旗标中点的模空间

Polymatroids and moduli of points in flags

本文引入并研究了仿射空间旗标中加权标记点的模空间的不同紧化，利用加权广义 Fulton-MacPherson 紧化构造，并证明了某些权重下为环面紧化，同构于 polymatroid 相关的簇，且具有 Losev-Manin 纤维化结构。

重新审视 $\mathbb{P}^1$ 上 8 点模空间

Revisiting the moduli space of 8 points on $\mathbb{P}^1$

本文证明 $\mathbb{P}^1$ 上 8 点模空间的 Deligne-Mostow 同构不能提升为 Kirwan 爆开与环面紧化间的态射，且两空间非 $K$-等价，并借助 Fulton-MacPherson 紧化揭示其与极小模型纲领的联系。

图配置空间、合同腺与形式性

Graphical configuration spaces, Contractads and Formality

本文研究图配置空间的合同腺结构，证明其在特定条件下形式性，并探讨了循环图的共形式性障碍及二维合同腺的细胞结构。