Smooth Compactification Research Articles

On the distribution of points of bounded height on equivariant compactifications of vector groups

We prove asymptotic formulas for the number of rational points of bounded height on smooth equivariant compactifications of the affine space.

A Smooth Space of Tetrahedra

We construct a smooth symmetric compactification of the space of all labeled tetrahedra in P3.

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 Sn manifest at each stage. The strata of the normal crossing divisor at infinity are labeled by leveled trees and their structure is studied. This is the maximal wonderful compactification in the sense of De Concini–Procesi, and it has a strata-compatible surjection onto the Fulton–MacPherson compactification. The degenerate configurations added in the compactification are geometrically described by polyscreens similar to the screens of Fulton and MacPherson. In characteristic 0, isotropy subgroups of the action of Sn on X〈n〉 are abelian, thus X〈n〉 may be a step toward an explicit resolution of singularities of the symmetric products X/Sn.

Twisted cubics, bis

We consider the smooth compactification constructed in [12] for a space of varieties like twisted cubics. We show this compactification embeds naturally in a product of flag varieties.

On the Brauer group

For an arithmetic model of a Fermat surface or a hyperkahler variety with Betti number over a purely imaginary number field , we prove the finiteness of the -components of for all primes . This yields a variant of a conjecture of M. Artin.If is a smooth projective irregular surface over a number field and , then the -primary component of is an infinite group for every prime . Let be the universal family of elliptic curves with a Jacobian structure of level over a number field . Assume that . If is a smooth projective compactification of the surface , then the -primary component of is a finite group for each sufficiently large prime .

GEOMETRY OF THE MODULI OF HIGHER SPIN CURVES

This article treats various aspects of the geometry of the moduli [Formula: see text] of r-spin curves and its compactification [Formula: see text]. Generalized spin curves, or r-spin curves, are a natural generalization of 2-spin curves (algebraic curves with a theta-characteristic), and have been of interest lately because of the similarities between the intersection theory of these moduli spaces and that of the moduli of stable maps. In particular, these spaces are the subject of a remarkable conjecture of E. Witten relating their intersection theory to the Gelfand–Dikii (KdVr) heirarchy. There is also a W-algebra conjecture for these spaces [16] generalizing the Virasoro conjecture of quantum cohomology. For any line bundle [Formula: see text] on the universal curve over the stack of stable curves, there is a smooth stack [Formula: see text] of triples (X, ℒ, b) of a smooth curve X, a line bundle ℒ on X, and an isomorphism [Formula: see text]. In the special case that [Formula: see text] is the relative dualizing sheaf, then [Formula: see text] is the stack [Formula: see text] of r-spin curves. We construct a smooth compactification [Formula: see text] of the stack [Formula: see text], describe the geometric meaning of its points, and prove that its coarse moduli is projective. We also prove that when r is odd and g>1, the compactified stack of spin curves [Formula: see text] and its coarse moduli space [Formula: see text] are irreducible, and when r is even and [Formula: see text] is the disjoint union of two irreducible components. We give similar results for n-pointed spin curves, as required for Witten's conjecture, and also generalize to the n-pointed case the classical fact that when [Formula: see text] is the disjoint union of d(r) components, where d(r) is the number of positive divisors of r. These irreducibility properties are important in the study of the Picard group of [Formula: see text] [15], and also in the study of the cohomological field theory related to Witten's conjecture [16, 34].

Higher-dimensional ball quotients and the invariant quintic

We prove a conjecture made earlier concerning a beautiful algebraic fourfold, a quintic in projective five-space invariant under the Weyl group of typeE 6, to the effect that a certain birational model of this variety is a smooth compactification of a ball quotient. To prove this, we first state and prove a general result which gives a criterion for checking whether a variety of dimensionN≥3 is a (compactification of a) ball quotient. We then go on to identify the group up to commensurability class.

Zero-cycles on Hilbert-Blumenthal surfaces

Introduction. The main object of study in this paper with respect to zero-cycles is a special class of Hilbert-Blumenthal surfaces X, which are defined over Q as smooth compactifications of quasi-projective varieties S/Q, more precisely, of coarse moduli schemes S that represent the moduli stack of polarized abelian surfaces with real multiplication by the ring of integers in a real quadratic field F = Q( √ d). We assume that d = q is a prime ≡ 1(4) and that the class number of F is 1. Then S(C), the complex points of S, can be described as H×H/ SL2(OF ), where H is the upper halfplane. In the early seventies, Hirzebruch and Zagier [HZ] defined for each integer N a curve TN on S (called Hirzebruch-Zagier cycles) and showed that their intersection numbers occur as Fourier coefficients of modular forms of level q with Nebentypes eq , the quadratic character of F/ Q. In this connection with modular forms, HirzebruchZagier cycles reveal very similar properties to Hecke correspondences on the selfproduct of the modular curve X0(q). This crucial observation of Hirzebruch and Zagier, together with Tunnell’s proof of the Tate conjecture for a product of modular curves, inspired Harder, Langlands, and Rapoport [HLR] to prove the Tate conjecture for divisors on Hilbert-Blumenthal surfaces over abelian number fields. (The proof of the Tate conjecture was then accomplished by Klingenberg [Kl] and Murty and Ramakrishnan [MR] in the general case.) From the new strategy to study torsion zero-cycles on algebraic surfaces as developed in [LS] and used in [L1], [L2] (compare also [LR]), it is clear that a crucial point is the Tate conjecture in characteristic p at good reduction primes. As one of our main results, we prove the Tate conjecture in characteristic p for primes p that split in F for a certain class of Hilbert-Blumenthal surfaces. For this we recall that to each modular cusp form f of weight 2, level q, and Nebentypes eq , that is, f ∈ S2(� 0(q), eq ) ,w e associate a Hilbert modular cusp form ˆ f ∈ S2(SL2(OF )) under the Doi-Naganuma

Algebraicity criteria for almost homogeneous complex spaces

We give some algebraicity criteria on almost homogeneous spaces and thereby generalize the following result of D. Luna: every smooth equivariant Moishezon compactification of an algebraic torus is in fact algebraic.

Formulas for the Unramified Brauer Group of a Principal Homogeneous Space of a Linear Algebraic Group

For a smooth compactification V of a principal homogeneous space E under a connected linear algebraic group G defined over a field k of characteristic zero, we present two formulas expressing BrV/Brk in terms of G.

The Space of Triangles, Vanishing Theorems, and Combinatorics

We consider compactifications of (Pn)3\\∪Δij, the space of triples of distinct points in projective 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 for the Schur modules; on the other, a combinatorial picture of the space of sections. Cohomology vanishing theorems play a key role in our analysis.

Canonical bundles of Deligne-Lusztig varieties

In this paper we consider Deligne–Lusztig varieties. We explicitly describe the canonical bundles of their smooth compactifications in terms of homogeneous line bundles pulled back from G/B. Using this description we show that the members (one member in each dimension) of a special family of Deligne–Lusztig varieties have ample canonical bundles. A consequence is that, unlike the closely related Schubert varieties, Deligne–Lusztig varieties are not in general Frobenius split. Several examples are given. Among these we exhibit (Example 4) an infinite family of counter-examples to the Miyaoka–Yau inequality (one for each prime power).

The Hodge polynomial of the Fulton-MacPherson compactification of configuration spaces

Let X be a smooth complex variety. In the paper "A compactification of configuration spaces," Fulton and MacPherson construct a smooth compactification X [ n ] of the configuration space of n distinct labelled points on X ; the space X [ n ] is projective if X is projective. In this paper, we obtain a generating function which expresses the virtual Hodge polynomials of the spaces X [ n ] in terms of the virtual Hodge polynomial of X . This enables us to read off the Euler characteristic of X [ n ] in terms of that of X and in the case when X is projective, the Hodge polynomial and the Poincaré polynomial of X [ n ] in terms of those of X .

On the comparison theorem for elementary irregular D-modules

Let U be a smooth quasi-projective variety over C and let f be a regular function on U. Let DU be the sheaf of algebraic differential operators on U and let M be a regular holonomic DU-module: here, regular means that there exists some smooth compactification X of U and some extension of M as a DX-module which is regular holonomic on X (one also may avoid the use of a smooth compactification to define regularity, see [17]).

Hecke correspondences and Betti cohomology groups for smooth compactifications of Hilbert modular varieties of dimension $\leq 3$

Hecke correspondences and Betti cohomology groups for smooth compactifications of Hilbert modular varieties of dimension $\leq 3$

An example of a non-projective smooth compactification of ?3 with second Betti number equal to one

An example of a non-projective smooth compactification of ?3 with second Betti number equal to one

A smooth compactification of the space of transfer functions with fixed McMillan degree

It is a classical result of Clark that the space of all proper or strictly properp ×m transfer functions of a fixed McMillan degreed has, in a natural way, the structure of a noncompact, smooth manifold. There is a natural embedding of this space into the set of allp × (m+p) autoregressive systems of degree at mostd. Extending the topology in a natural way we will show that this enlarged topological space is compact. Finally we describe a homogenization process which produces a smooth compactification.

Algebraic Cycles and the Hodge Structure of a Kuga Fiber Variety

Let $\tilde A$ denote a smooth compactification of the $k$-fold fiber product of the universal family ${A^1} \to M$ of elliptic curves with level $N$ structure. The purpose of this paper is to completely describe the algebraic cycles in and the Hodge structure of the Betti cohomology ${H^{\ast } }(\tilde A,\mathbb {Q})$ of $\tilde A$ , for by doing so we are able (a) to verify both the usual and generalized Hodge conjectures for $\tilde A$ ; (b) to describe both the kernel and the image of the Abel-Jacobi map from algebraic cycles algebraically equivalent to zero (modulo rational equivalence) into the Griffiths intermediate Jacobian; and (c) to verify Tate’s conjecture concerning the algebraic cycles in the étale cohomology $H_{{\text {et}}}^{\ast } (\tilde A \otimes \bar {\mathbb {Q}},{\mathbb {Q}_l})$. The methods used lead also to a complete description of the Hodge structure of the Betti cohomology ${H^{\ast } }({E^k},\mathbb {Q})$ of the $k$-fold product of an elliptic curve $E$ without complex multiplication, and a verification of the generalized Hodge conjecture for ${E^k}$ .

On the Deformations of Certain Moduli Spaces of Vector Bundles

Introduction. Let X be a smooth projective curve of genus g > 2 and Md(n, L) the moduli space of semi-stable bundles E of rank n and degree d, with det(E) z L. When (n, d) = 1 the variety Md(n, L) is smooth projective. For these varieties a theorem of Narasimhan and Ramanan [NR2] states that the number of moduli of Md(n, L) is the same as that of the curve X. The purpose of this paper is to prove this statement for the smooth compactification N, constructed by Seshadri in [S2], of Mo(2, Ox)s, (the moduli space of stable bundles, over a curve X of genus g > 5). More precisely, we prove the following

Algebraic cycles and the Hodge structure of a Kuga fiber variety

Let A ~ \tilde A denote a smooth compactification of the k k -fold fiber product of the universal family A 1 → M {A^1} \to M of elliptic curves with level N N structure. The purpose of this paper is to completely describe the algebraic cycles in and the Hodge structure of the Betti cohomology H ∗ ( A ~ , Q ) {H^{\ast } }(\tilde A,\mathbb {Q}) of A ~ \tilde A , for by doing so we are able (a) to verify both the usual and generalized Hodge conjectures for A ~ \tilde A ; (b) to describe both the kernel and the image of the Abel-Jacobi map from algebraic cycles algebraically equivalent to zero (modulo rational equivalence) into the Griffiths intermediate Jacobian; and (c) to verify Tate’s conjecture concerning the algebraic cycles in the étale cohomology H et ∗ ( A ~ ⊗ Q ¯ , Q l ) H_{{\text {et}}}^{\ast } (\tilde A \otimes \bar {\mathbb {Q}},{\mathbb {Q}_l}) . The methods used lead also to a complete description of the Hodge structure of the Betti cohomology H ∗ ( E k , Q ) {H^{\ast } }({E^k},\mathbb {Q}) of the k k -fold product of an elliptic curve E E without complex multiplication, and a verification of the generalized Hodge conjecture for E k {E^k} .