Local Equations for the Toric Hilbert Scheme

We obtain local equations for the toric Hilbert scheme, which parametrizes all ideals with the same multigraded Hilbert function as a given toric ideal. We also prove a conjecture of Sturmfels' providing a criterion for an ideal to have such a Hilbert function.     

A note on finite abelian gerbes over toric Deligne-Mumford stacks

Any toric Deligne-Mumford stack is a -gerbe over the underlying toric orbifold for a finite abelian group . In this paper we give a sufficient condition so that certain kinds of gerbes over a toric Deligne-Mumford stack are again toric Deligne-Mumford stacks.

     

Toric Rigid Spaces

This paper gives a method to construct rigid spaces, which is similar to the method used to construct toric schemes.     

K-Theory of Algebraic Tori and Toric Varieties

It is proved that under certain conditions the group K _n (X) of a smooth projective variety X over a field F is a natural direct summand of K _n (A) for some separable F-algebra A. As an application we study the K-groups of toric models and toric varieties. A presentation in terms of generators and relations of the groupK ₀(T) for an algebraic torus T is given.     

Intersection Cohomology on Nonrational Polytopes⋆

We consider a fan as a ringed space (with finitely many points). We develop the corresponding sheaf theory and functors, such as direct image R_* ( is a subdivision of a fan), Verdier duality, etc. The distinguished sheaf , called the minimal sheaf plays the role of an equivariant intersection cohomology complex on the corresponding toric variety (which exists if is rational). Using we define the intersection cohomology space IH(). It is conjectured that a strictly convex piecewise linear function on acts as a Lefschetz operator on IH(). We show that this conjecture implies Stanley's conjecture on the unimodality of the generalized h-vector of a convex polytope.     

The toric cobordisms

We introduce the notions of oriented and unoriented cobordisms in the class of closed 3-manifolds fibered by tori and compute the corresponding cobordism groups.

     

Generic lattice ideals

A concept of genericity is introduced for lattice ideals (and hence for ideals defining toric varieties) which ensures nicely structured homological behavior. For a generic lattice ideal we construct its minimal free resolution and we show that it is induced from the Scarf resolution of any reverse lexicographic initial ideal.

     

Values of zeta functions at negative integers, Dedekind sums and toric geometry

We study relations among special values of zeta functions, invariants of toric varieties, and generalized Dedekind sums. In particular, we use invariants arising in the Todd class of a toric variety to give a new explicit formula for the values of the zeta function of a real quadratic field at nonpositive integers. We also express these invariants in terms of the generalized Dedekind sums studied previously by several authors. The paper includes conceptual proofs of these relations and explicit computations of the various zeta values and Dedekind sums involved.

     

The set of toric minimal log discrepancies

We describe the set of minimal log discrepancies of toric log varieties, and study its accumulation points. This work is supported by a 21st Century COE Kyoto Mathematics Fellowship, and a JSPS Grant-in-Aid No 17740011.     

Asymptotically Maximal Families of Hypersurfaces in Toric Varieties

A real algebraic variety is maximal (with respect to the Smith-Thom inequality) if the sum of the Betti numbers (with coefficients) of the real part of the variety is equal to the sum of Betti numbers of its complex part. We prove that there exist polytopes that are not Newton polytopes of any maximal hypersurface in the corresponding toric variety. On the other hand we show that for any polytope Δ there are families of hypersurfaces with the Newton polytopes that are asymptotically maximal when λ tends to infinity. We also show that these results generalize to complete intersections.     

On the Chiral Ring of Calabi–Yau Hypersurfaces in Toric Varieties

This paper is devoted to the calculation of the B-model chiral ring, used in physics, for semiample Calabi–Yau hypersurfaces. We also study the cohomology of semiample hypersurfaces.     

Strong Exceptional Sequences Provided by Quivers

Let Q be a finite quiver without oriented cycles. Denote by U (Q) the fine moduli space of stable thin sincere representations of Q with respect to the canonical stability notion. We prove Ext _(Q) ^l (U, U) = 0 for all l > 0 and compute the endomorphism algebra of the universal bundle U. Moreover, we obtain a necessary and sufficient condition for when this algebra is isomorphic to the path algebra of the quiver Q. If so, then the bounded derived categories of finitely generated right k Q-modules and that of coherent sheaves on (Q) are related via the full and faithful functor – _kQ ^L U.     

Remarks on toric manifolds whose Chern characters are positive

Abstract

We show that the second Chern character of any projective toric manifold of Picard number three is not positive. In connection with this result, we give various examples of the positivity of higher Chern characters of projective toric manifolds.

Communicated by Stephen L. Kleiman     

Wedge Operations and Doubling Operations of Real Toric Manifolds

This paper deals with two things.First,the cohomology of canonical extensions of real topological toric manifolds is computed when coefficient ring G is a commutative ring in which 2 is unit in G.Second,the author focuses on a specific canonical extensions called doublings and presents their various properties.They include existence of infinitely many real topological toric manifolds admitting complex structures,and a way to construct infinitely many real toric manifolds which have an odd torsion in their cohomology groups.Moreover,some questions about real topological toric manifolds related to Halperin's toral rank conjecture are presented.     

Generators for semigroup of Lipman

There is a correspondence between functions in the maximal ideal of the local ring of a rational singularity and certain positive divisors supported on the exceptional fiber of a resolution of the singularity. Here we give an algorithm to obtain a generating set over ℤ of these divisors.     

Bisoptic curves of hyperbolas

Given a hyperbola, we study its bisoptic curves, i.e. the geometric locus of points through which passes a pair of tangents making a fixed angle θ or 180° ? θ. This question has been addressed in a previous paper for parabolas and for ellipses, showing hyperbolas and spiric curves, respectively. Here the requested geometric locus can be empty. If not, it is a punctured spiric curve, and two cases occur: the curve can have either one loop or two loops. Finally, we reconstruct explicitly the spiric curve as the intersection of a plane with a self-intersecting torus.     

Small Cover的L-S畴数

我们证明了如下结论:任意n维small cover的Lusternik-Schnirelmann畴数等于n;任意2n维toric-流形的Lusternik-Schnirelmann畴数等于n.我们的结果依赖于如下两个事实,一个是不等式cup(M) ≤ cat(M) ≤dim(M)/r,它导致我们去计算cup(M).另一...     

Closed Geodesics on Incomplete Surfaces

We consider the problem of finding embedded closed geodesics on the two-sphere with an incomplete metric defined outside a point. Various techniques including curve shortening methods are used The authors acknowledge the support of the Australian Research Council     

Jet schemes and minimal toric embedded resolutions of rational double point singularities

Using the structure of the jet schemes of rational double point singularities, we construct “minimal embedded toric resolutions” of these singularities. We also establish, for these singularities, a correspondence between a natural class of irreducible components of the jet schemes centered at the singular locus and the set of divisors which appear on every “minimal embedded toric resolution”. We prove that this correspondence is bijective except for the E₈ singulartiy. This can be thought as an embedded Nash correspondence for rational double point singularities.