Reducibility of Punctual Hilbert Schemes of Cone Varieties

In this short note we show that for any pair of positive integers (d, n) with n > 2 and d > 1 or n = 2 and d > 4, there always exist projective varieties X ? ? ^N of dimension n and degree d and an integer s ₀ such that Hilb ^s (X) is reducible for all s ≥ s ₀. X will be a projective cone in ? ^N over an arbitrary projective variety Y ? ? ^N?1. In particular, we show that, opposite to the case of smooth surfaces, there exist projective surfaces with a single isolated singularity which have reducible Hilbert scheme of points.     

Reduction criteria for modules

RÉSUMÉ

On vérifie, en degré d ≤ 11, la conjecture suivante: “le schéma de Hilbert des courbes lisses connexes de degré d et de genre g de l'espace projectif á trois dimensions sur le corps des complexes est irréductible pour g ≤ 2d ? 9”

ABSTRACT

We check that the following conjecture holds true for d ≤ 11 : “the Hilbert scheme of smooth and connected space curves of degree d and genus g is irreducible provided that g ≤ 2d ? 9.”     

Hilbert Scheme of a Pair of Codimension Two Linear Subspaces

We study the component H _n of the Hilbert scheme whose general point parameterizes a pair of codimension two linear subspaces in ? ⁿ for n ≥ 3. We show that H _n is smooth and isomorphic to the blow-up of the symmetric square of 𝔾(n ? 2, n) along the diagonal. Further H _n intersects only one other component in the full Hilbert scheme, transversely. We determine the stable base locus decomposition of its effective cone and give modular interpretations of the corresponding models, hence conclude that H _n is a Mori dream space.     

On the Non-Splitting of the Normal Bundle Sequence

In a recent article, Paltin Ionescu and Flavia Repetto proved that if X ? ? ⁿ is a smooth projective variety over ? such that its normal bundle sequence splits over some curve C ? X, then X a linear subspace in ? ⁿ . In this note, we give a purely geometric proof of this result that is valid in arbitrary characteristic.     

Real Hypersurfaces Having Many Pseudo-hyperplanes

Abstract

Let nand dbe natural integers satisfying n ≥ 3 and d ≥ 10. Let Xbe an irreducible real hypersurface Xin ? ⁿ of degree dhaving many pseudo-hyperplanes. Suppose that Xis not a projective cone. We show that the arrangement ? of all d ? 2 pseudo-hyperplanes of Xis trivial, i.e., there is a real projective linear subspace Lof ? ⁿ (?) of dimension n ? 2 such that L ? Hfor all H ∈ ?. As a consequence, the normalization of Xis fibered over ?¹in quadrics. Both statements are in sharp contrast with the case n = 2; the first statement also shows that there is no Brusotti-type result for hypersurfaces in ? ⁿ , for n ≥ 3.     

Localisations of non prime e-vh-orders

Abstract

If A is an n × n complex matrix and x ∈ ? ⁿ , the conjecture is that if we take the kth power of each component of Ax, the resulting vector belongs to the range of the matrix obtained by taking the kth power of the entries of AA ^?, where A ^? is the adjoint of A. The conjecture is proved here for any k ≥ 2 when we add assumptions of either low dimension (namely, n ≤ 4) or low corank (0, 1, and, with some technical restrictions, 2). This problem arises in the study of the Jacobian Conjecture.     

Threefolds in ℙ5 with One Apparent Quadruple Point

Abstract

In this article we classify all the smooth threefolds in ?⁵ with an apparent quadruple point provided that the family of its 4-secant lines is an irreducible (first order) congruence. This is sufficient to conclude the classification of all the smooth codimension two varieties in ? ⁿ with one apparent (n ? 1)-point and with irreducible family of (n ? 1)-secant lines.     

Low Frequency Estimates and Local Energy Decay for Asymptotically Euclidean Laplacians

For Riemannian metrics G on ? ^d which are long range perturbations of the flat one, we prove estimates for (? Δ _G  ? λ ?iε)^?n as λ → 0, which are uniform with respect to ε, for all n ≤ [d/2] +1 in odd dimension and n ≤ d/2 in even dimension. We also give applications to the time decay of Schrödinger and Wave (or Klein–Gordon) equations.     

Inverse Problems for Time-Dependent Singular Heat Conductivities: Multi-Dimensional Case

We consider an inverse boundary value problem for the heat equation ? _t u = div (γ? _x u) in (0, T) × Ω, u = f on (0, T) × ?Ω, u| _t=0 = u ₀, in a bounded domain Ω ? ? ⁿ , n ≥ 2, where the heat conductivity γ(t, x) is piecewise constant and the surface of discontinuity depends on time: γ(t, x) = k ² (x ∈ D(t)), γ(t, x) = 1 (x ∈ Ω?D(t)). Fix a direction e* ∈ 𝕊 ^n?1 arbitrarily. Assuming that ?D(t) is strictly convex for 0 ≤ t ≤ T, we show that k and sup {e*·x; x ∈ D(t)} (0 ≤ t ≤ T), in particular D(t) itself, are determined from the Dirichlet-to-Neumann map : f → ?_ν u(t, x)|_{(0, T)×?Ω}. The knowledge of the initial data u ₀ is not used in the proof. If we know min_0≤t≤T (sup _x∈D(t) x·e*), we have the same conclusion from the local Dirichlet-to-Neumann map. Numerical examples of stationary and moving circles inside the unit disk are shown. The results have applications to nondestructive testing. Consider a physical body consisting of homogeneous material with constant heat conductivity except for a moving inclusion with different conductivity. Then the location and shape of the inclusion can be monitored from temperature and heat flux measurements performed at the boundary of the body. Such a situation appears for example in blast furnaces used in ironmaking.     

Lattices Generated by Orbits of Flats Under Finite Affine Groups

Let AG(n, _q ) be the n-dimensional affine space over  _q . For a given integer m with 0 ≤ m ≤ n, all m-flats form an orbit, denoted by ?(m,n), under the action of the affine group AGL _n ( _q ) of AG(n, _q ). Denote the set of all intersections of m-flats in ?(m,n) by ?(m,n). By ordering ?(m,n) by ordinary or reverse inclusion, two classes of lattices are obtained. This article discusses their geometricity, and computes their character polynomials.     

Ample Vector Bundles with Small g − q

Let X be a smooth complex projective variety and let Z ? X be a smooth submanifold of dimension ≥ 2, which is the zero locus of a section of an ample vector bundle ? of rank dim X ? dim Z ≥ 2 on X. Let H be an ample line bundle on X, whose restriction H _Z to Z is generated by global sections. The structure of triplets (X,?,H) as above is described under the assumption that the curve genus of the corank-1 vector bundle ? ⊕ H ^{⊕ (dim Z?1)} is ≤ h ¹( _X ) + 2.     

Weak Global Dimension of Coherent Rings

In this article, we study the weak global dimension of coherent rings in terms of the left FP-injective resolutions of modules. Let R be a left coherent ring and ? ? the class of all FP-injective left R-modules. It is shown that wD(R) ≤ n (n ≥ 1) if and only if every nth ? ?-syzygy of a left R-module is FP-injective; and wD(R) ≤ n (n ≥ 2) if and only if every (n ? 2)th ? ?-syzygy in a minimal ? ?-resolution of a left R-module has an FP-injective cover with the unique mapping property. Some results for the weak global dimension of commutative coherent rings are also given.     

Inverse Spectral Problems in Rectangular Domains

We consider the Schrödinger operator ? Δ + q in domains of the form R = {x ∈ ? ⁿ : 0 ≤ x _i  ≤ a _i , i = 1,…, n} with either Dirichlet or Neumann boundary conditions on the faces of R, and study the constraints on q imposed by fixing the spectrum of ? Δ + q with these boundary conditions. We work in the space of potentials, q, which become real-analytic on ? ⁿ when they are extended evenly across the coordinate planes and then periodically. Our results have the corollary that there are no continuous isospectral deformations for these operators within that class of potentials. This work is based on new formulas for the trace of the wave group in this setting. In addition to the inverse spectral results these formulas lead to asymptotic expansions for the traces of the wave and heat kernels on rectangular domains.     

Maximal Subgroups of Finite Orthogonal Groups Stabilizing Spreads of Lines

We prove that, for q odd and n ≥ 3, the group G = O _n (q ²) · 2 is maximal in either the orthogonal group O _2n (q) or the special orthogonal group SO _2n (q). The group G corresponds to the stabilizer of a spread of lines of PG(2n ? 1, q) in which some lines lie on a quadric, some are secant to the quadric, and others are external to the quadric.     

The Cauchy Problem for Defocusing Nonlinear Schrödinger Equations with Non-Vanishing Initial Data at Infinity

For rather general nonlinearities, we prove that defocusing nonlinear Schrödinger equations in ? ⁿ (n ≤ 4), with non-vanishing initial data at infinity u ₀, are globally well-posed in u ₀ + H ¹. The same result holds in an exterior domain in ? ⁿ , n = 2, 3.     

Vanishing Theorems for Conformally Compact Manifolds

Abstract

We study L ² harmonic p-forms on conformally compact manifolds with a rather weak boundary regularity assumption. We proved that if the lower bound of the curvature operator is great than or equal to ?1 and the infimum of the L ² spectrum of the Laplacian great than p(n ? p) for some p ≤ n/2, then there is no nontrivial L ² harmonic p-form.     

Prophet Regions for Independent [0, 1]-Valued Random Variables with Random Discounting

Abstract

Let X ₁, X ₂… and B ₁, B ₂… be mutually independent [0, 1]-valued random variables, with EB _j  = β > 0 for all j. Let Y _j  = B ₁ … sB _j?1 X _j for j ≥ 1. A complete comparison is made between the optimal stopping value V(Y ₁,…,Y _n ):=sup{EY _τ:τ is a stopping rule for Y ₁,…,Y _n } and E(max _1≤j≤n Y _j ). It is shown that the set of ordered pairs {(x, y):x = V(Y ₁,…,Y _n ), y = E(max _1≤j≤n Y _j ) for some sequence Y ₁,…,Y _n obtained as described} is precisely the set {(x, y):0 ≤ x ≤ 1, x ≤ y ≤ Ψ _{n, β}(x)}, where Ψ _{n, β}(x) = [(1 ? β)n + 2β]x ? β^?(n?2) x ² if x ≤ β ^n?1, and Ψ _{n, β}(x) = min _j≥1{(1 ? β)jx + β ^j } otherwise. Sharp difference and ratio prophet inequalities are derived from this result, and an analogous comparison for infinite sequences is obtained.     

A Generalization of a Lemma of Sullivan

Consider an irreducible polynomial of the form f(X) = X ^p  ? aX ? b ∈ 𝔽[X] and α a root of f(X), where 𝔽 is a field of characteristic p. In 1975, F.J. Sullivan stated a lemma that provides the trace, taken with respect to the extension 𝔽(α)/𝔽, of elements of the form α ⁿ , where 0 ≤ n ≤ p ² ? 1. We present a generalization of Sullivan's Lemma and provide another proof of the original lemma. We explain how computing Tr(α ⁿ ) for n < p ^r can be reduced to computing the traces Tr(α ^m ) for all m ≤ r(p ? 1).