Linear Precision for Toric Surface Patches

We classify the homogeneous polynomials in three variables whose toric polar linear system defines a Cremona transformation. This classification includes, as a proper subset, the classification of toric surface patches from geometric modeling which have linear precision. Besides the well-known tensor product patches and Bézier triangles, we identify a family of toric patches with trapezoidal shape, each of which has linear precision. Furthermore, Bézier triangles and tensor product patches are special cases of trapezoidal patches.     

On equivalences of derived and singular categories

Let X and Y be two smooth Deligne-Mumford stacks and consider a pair of functions f: X → $ \mathbb{A}^1 $ \mathbb{A}^1 , g:Y → $ \mathbb{A}^1 $ \mathbb{A}^1 . Assuming that there exists a complex of sheaves on X × $ \mathbb{A}^1 $ \mathbb{A}^1 Y which induces an equivalence of D ^b (X) and D ^b (Y), we show that there is also an equivalence of the singular derived categories of the fibers f ⁻¹(0) and g ⁻¹(0). We apply this statement in the setting of McKay correspondence, and generalize a theorem of Orlov on the derived category of a Calabi-Yau hypersurface in a weighted projective space, to products of Calabi-Yau hypersurfaces in simplicial toric varieties with nef anticanonical class.     

On the classification of toric Fano 4-folds

The biregular classification of smoothd-dimensional toric Fano varieties is equivalent to the classification of special simplicial polyhedraP in ℝ ^d , the so-called Fano polyhedra, up to an isomorphism of the standard lattice . In this paper, we explain the complete biregular classification of all 4-dimensional smooth toric Fano varieties. The main result states that there exist exactly 123 different types of toric Fano 4-folds somorphism. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory Vol. 56. Algebraic Geometry-9, 1998.     

Analytic criterion for linear convexity of Hartogs domains with smooth boundary in $ {\mathbb{H}^2} $

We establish a criterion for the local linear convexity of sets in the two-dimensional quaternion space \mathbbH² {\mathbb{H}^2} that are analogs of bounded Hartogs domains with smooth boundary in the two-dimensional complex space \mathbbC² {\mathbb{C}^2} .     

Combinatorics of the Toric Hilbert Scheme

The toric Hilbert scheme is a parameter space for all ideals with the same multigraded Hilbert function as a given toric ideal. Unlike the classical Hilbert scheme, it is unknown whether toric Hilbert schemes are connected. We construct a graph on all the monomial ideals on the scheme, called the flip graph, and prove that the toric Hilbert scheme is connected if and only if the flip graph is connected. These graphs are used to exhibit curves in P ⁴ whose associated toric Hilbert schemes have arbitrary dimension. We show that the flip graph maps into the Baues graph of all triangulations of the point configuration defining the toric ideal. Inspired by the recent discovery of a disconnected Baues graph, we close with results that suggest the existence of a disconnected flip graph and hence a disconnected toric Hilbert scheme. Received May 15, 2000, and in revised form March 8, 2001. Online publication January 7, 2002.     

Constructions in Sasakian geometry

We first generalize the join construction described previously by the first two authors [4] for quasi-regular Sasakian-Einstein orbifolds to the general quasi-regular Sasakian case. This allows for the further construction of specific types of Sasakian structures that are preserved under the join operation, such as positive, negative, or null Sasakian structures, as well as Sasakian-Einstein structures. In particular, we show that there are families of Sasakian-Einstein structures on certain 7-manifolds homeomorphic to S ² × S ⁵. We next show how the join construction emerges as a special case of Lerman’s contact fibre bundle construction [32]. In particular, when both the base and the fiber of the contact fiber bundle are toric we show that the construction yields a new toric Sasakian manifold. Finally, we study toric Sasakian manifolds in dimension 5 and show that any simply-connected compact oriented 5-manifold with vanishing torsion admits regular toric Sasakian structures. This is accomplished by explicitly constructing circle bundles over the equivariant blow-ups of Hirzebruch surfaces. During the preparation of this work the first two authors were partially supported by NSF grants DMS-0203219 and DMS-0504367.     

Bicomplex hyperfunctions

In this paper, we consider bicomplex holomorphic functions of several variables in _boxclose C^n{{\mathbb B}{\mathbb C}^n} .We use the sheaf of these functions to define and study hyperfunctions as their relative 3n-cohomology classes. We show that such hyperfunctions are supported by the Euclidean space \mathbb Rⁿ{{\mathbb R}^n} within the bicomplex space \mathbb B\mathbb Cⁿ{{\mathbb B}{\mathbb C}^n}, and we construct an abstract Dolbeault complex that provides a fine resolution for the sheaves of bicomplex holomorphic functions. As a corollary, we show how that the bicomplex hyperfunctions can be represented as classes of differential forms of degree 3n − 1.     

Σ-definability of uncountable models of <Emphasis Type="Italic">c</Emphasis>-simple theories

We show that each c-simple theory with an additional discreteness condition has an uncountable model Σ-definable in ℍ$ \mathbb{H} $ \mathbb{H} ($ \mathbb{L} $ \mathbb{L} ), where $ \mathbb{L} $ \mathbb{L} is a dense linear order. From this we establish the same for all c-simple theories of finite signature that are submodel complete.     

Prime filtrations and Stanley decompositions of squarefree modules and Alexander duality

In this paper we study how prime filtrations and squarefree Stanley decompositions of squarefree modules over the polynomial ring and over the exterior algebra behave with respect to Alexander duality. The results which we obtained suggest a lower bound for the regularity of a \mathbb Zⁿ{\mathbb {Z}^n}-graded module in terms of its Stanley decompositions. For squarefree modules this conjectured bound is a direct consequence of Stanley’s conjecture on Stanley decompositions. We show that for pretty clean rings of the form R/I, where I is a monomial ideal, and for monomial ideals with linear quotient our conjecture holds.     

Covering and packing in {\mathbb Z^n} and {\mathbb R^n}. (II)

In the present part (II) we will deal with the group \mathbb G = \mathbb Zⁿ{\mathbb G = \mathbb Z^n} , and we will study the effect of linear transformations on minimal covering and maximal packing densities of finite sets A ì \mathbb Zⁿ{\mathcal A \subset {\mathbb Z}^n} . As a consequence, we will be able to show that the set of all densities for sets A{\mathcal A} of given cardinality is closed, and to characterize four-element sets A ì \mathbb Zⁿ{\mathcal A \subset {\mathbb Z}^n} which are “tiles”. The present work will be largely independent of the first part (I) presented in [4].     

The Behaviors of Expansion Functor on Monomial Ideals and Toric Rings

In this article, we study some algebraic and combinatorial behaviors of expansion functor. We show that on monomial ideals some properties like polymatroidalness, weakly polymatroidalness, and having linear quotients are preserved under taking the expansion functor.

The main part of the article is devoted to study of toric ideals associated to the expansion of subsets of monomials which are minimal with respect to divisibility. It is shown that, for a given discrete polymatroid P, if toric ideal of P is generated by double swaps, then toric ideal of any expansion of P has such a property. This result, in a special case, says that White's conjecture is preserved under taking the expansion functor. Finally, the construction of Gröbner bases and some homological properties of toric ideals associated to expansions of subsets of monomials is investigated.     

Welschinger invariants of toric Del Pezzo surfaces with nonstandard real structures

The Welschinger invariants of real rational algebraic surfaces are natural analogs of the Gromov-Witten invariants, and they estimate from below the number of real rational curves passing through prescribed configurations of points. We establish a tropical formula for the Welschinger invariants of four toric Del Pezzo surfaces equipped with a nonstandard real structure. Such a formula for real toric Del Pezzo surfaces with a standard real structure (i.e., naturally compatible with the toric structure) was established by Mikhalkin and the author. As a consequence we prove that for any real ample divisor D on a surface Σ under consideration, through any generic configuration of c ₁(Σ)D − 1 generic real points, there passes a real rational curve belonging to the linear system |D|. To Vladimir Igorevich Arnold on the occasion of his 70th birthday     

Toric Aspects of the First Eigenvalue

In this paper we study the smallest non-zero eigenvalue \(\lambda _1\) of the Laplacian on toric Kähler manifolds. We find an explicit upper bound for \(\lambda _1\) in terms of moment polytope data. We show that this bound can only be attained for \(\mathbb C\mathbb P^n\) endowed with the Fubini–Study metric and therefore \(\mathbb C\mathbb P^n\) endowed with the Fubini–Study metric is spectrally determined among all toric Kähler metrics. We also study the equivariant counterpart of \(\lambda _1\) which we denote by \(\lambda _1^T\). It is the smallest non-zero eigenvalue of the Laplacian restricted to torus-invariant functions. We prove that \(\lambda _1^T\) is not bounded among toric Kähler metrics thus generalizing a result of Abreu–Freitas on \(S^2\). In particular, \(\lambda _1^T\) and \(\lambda _1\) do not coincide in general.     

On weighted spaces of holomorphic functions of several variables

We generalize the results of [11] and [12] for the unit ball $ \mathbb{B}_d $ \mathbb{B}_d of ℂ ^d . In particular, we show that under the weight condition (B) the weighted H ^∞-space on $ \mathbb{B}_d $ \mathbb{B}_d is isomorphic to ℓ^∞ and thus complemented in the corresponding weighted L ^∞-space. We construct concrete, generalized Bergman projections accordingly. We also consider the case where the domain is the entire space ℂ ^d . In addition, we show that for the polydisc $ \mathbb{D}^d $ \mathbb{D}^d ^d , the weighted H ^∞-space is never isomorphic to ℓ^∞.     

An Inverse Theorem for the Uniformity Seminorms Associated with the Action of {{\mathbb {F}^{\infty}_{p}}}

Let ${\mathbb {F}}Let \mathbb F{\mathbb {F}} a finite field. We show that the universal characteristic factor for the Gowers–Host–Kra uniformity seminorm U ^k (X) for an ergodic action (T_g)_{g ? \mathbb F^w}{(T_{g})_{{g} \in \mathbb {F}^{\omega}}} of the infinite abelian group \mathbb F^w{\mathbb {F}^{\omega}} on a probability space X = (X, B, m){X = (X, \mathcal {B}, \mu)} is generated by phase polynomials f: X ? S¹{\phi : X \to S^{1}} of degree less than C(k) on X, where C(k) depends only on k. In the case where k ￡ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} we obtain the sharp result C(k) = k. This is a finite field counterpart of an analogous result for \mathbb Z{\mathbb {Z}} by Host and Kra [HK]. In a companion paper [TZ] to this paper, we shall combine this result with a correspondence principle to establish the inverse theorem for the Gowers norm in finite fields in the high characteristic case k ￡ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} , with a partial result in low characteristic.     

A case of the dynamical Mordell–Lang conjecture

We prove a special case of a dynamical analogue of the classical Mordell–Lang conjecture. Specifically, let φ be a rational function with no periodic critical points other than those that are totally invariant, and consider the diagonal action of φ on (\mathbb P¹)^g{(\mathbb P^1)^g}. If the coefficients of φ are algebraic, we show that the orbit of a point outside the union of the proper preperiodic subvarieties of (\mathbb P¹)^g{(\mathbb P^1)^g} has only finite intersection with any curve contained in (\mathbb P¹)^g{(\mathbb P^1)^g}. We also show that our result holds for indecomposable polynomials φ with coefficients in \mathbb C{\mathbb C}. Our proof uses results from p-adic dynamics together with an integrality argument. The extension to polynomials defined over \mathbb C{\mathbb C} uses the method of specialization coupled with some new results of Medvedev and Scanlon for describing the periodic plane curves under the action of (φ, φ) on \mathbb A²{\mathbb A^2}.     

Hardy–Littlewood Inequalities for Fractional Derivatives of Invariant Harmonic Functions

We establish Hardy–Littlewood inequalities for fractional derivatives of M?bius invariant harmonic functions over the unit ball of \mathbb Rⁿ{\mathbb R^n} in mixed-norm spaces. In doing so we introduce a new criteria for the boundedness of operators in mixed-norm L ^p -spaces in terms of hyperbolic geometry of the real unit ball.     

The Hermite Functions Related to Infinite Series of Generalized Convolutions and Applications

In this paper, we show that arbitrary Hermite function or appropriate linear combination of those functions is a weight-function of four explicit generalized convolutions for the Fourier cosine and sine transforms. With respect to applications, normed rings on L¹(\mathbbR^d){L^1(\mathbb{R}^d)} are constructed, and sufficient and necessary conditions for the solvability and explicit solutions in L¹(\mathbbR^d){L^1(\mathbb{R}^d)} of the integral equations of convolution type are provided by using the constructed convolutions.