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1.
We show how to construct sparse polynomial systems that have non-trivial lower bounds on their numbers of real solutions. These are unmixed systems associated to certain polytopes. For the order polytope of a poset P this lower bound is the sign-imbalance of P and it holds if all maximal chains of P have length of the same parity. This theory also gives lower bounds in the real Schubert calculus through the sagbi degeneration of the Grassmannian to a toric variety, and thus recovers a result of Eremenko and Gabrielov. 相似文献
2.
Joaquín Moraga 《Journal of Pure and Applied Algebra》2019,223(8):3225-3237
In this note, we study linear systems on complete toric varieties X with an invariant point whose orbit under the action of contains the dense torus T of X. We give a characterization of such varieties in terms of its defining fan and introduce a new definition of expected dimension of linear systems which consider the contribution given by certain toric subvarieties. Finally, we study degenerations of linear systems on these toric varieties induced by toric degenerations. 相似文献
3.
Bernt Ivar Utstøl Nødland 《Journal of Pure and Applied Algebra》2018,222(3):508-533
We use Matsui and Takeuchi's formula for toric A-discriminants to give algorithms for computing local Euler obstructions and dual degrees of toric surfaces and 3-folds. In particular, we consider weighted projective spaces. As an application we give counterexamples to a conjecture by Matsui and Takeuchi. As another application we recover the well-known fact that the only defective normal toric surfaces are cones. 相似文献
4.
Let P be a simple lattice polytope. We define an action of the Hecke operators on E(P), the Ehrhart polynomial of P, and describe their effect on the coefficients of E(P). We also describe how the Brion–Vergne formula for E(P) transforms under the Hecke operators for nonsingular lattice polytopes P.
相似文献
5.
We show that any smooth Q-normal lattice polytope P of dimension n and degree d is a strict Cayley polytope if n?2d+1. This gives a sharp answer, for this class of polytopes, to a question raised by V.V. Batyrev and B. Nill. 相似文献
6.
In this paper, we give an algebro-geometric characterization of Cayley polytopes. As a special case, we also characterize lattice polytopes with lattice width one by using Seshadri constants. 相似文献
7.
We prove that any affine, resp. polarized projective, spherical variety admits a flat degeneration to an affine, resp. polarized projective, toric variety. Motivated by mirror symmetry, we give conditions for the limit toric variety to be a Gorenstein Fano, and provide many examples. We also provide an explanation for the limits as boundary points of the moduli space of stable pairs whose existence is predicted by the Minimal Model Program. 相似文献
8.
Leovigildo Alonso Tarrío Ana Jeremías López 《Journal of Pure and Applied Algebra》2009,213(7):1373-1398
We continue our study on infinitesimal lifting properties of maps between locally noetherian formal schemes started in [L. Alonso Tarrío, A. Jeremías López, M. Pérez Rodríguez, Infinitesimal lifting and Jacobi criterion for smoothness on formal schemes, Comm. Alg. 35 (2007) 1341-1367]. In this paper, we focus on some properties which arise specifically in the formal context. In this vein, we make a detailed study of the relationship between the infinitesimal lifting properties of a morphism of formal schemes and those of the corresponding maps of usual schemes associated to the directed systems that define the corresponding formal schemes. Among our main results, we obtain the characterization of completion morphisms as pseudo-closed immersions that are flat. Also, the local structure of smooth and étale morphisms between locally noetherian formal schemes is described: the former factors locally as a completion morphism followed by a smooth adic morphism and the latter as a completion morphism followed by an étale adic morphism. 相似文献
9.
Toric varieties,lattice points and Dedekind sums 总被引:8,自引:0,他引:8
James E. Pommersheim 《Mathematische Annalen》1993,295(1):1-24
10.
For each dimension d, d-dimensional integral simplices with exactly one interior integral point have bounded volume. This was first shown by Hensley. Explicit volume bounds were determined by Hensley, Lagarias and Ziegler, Pikhurko, and Averkov. In this paper we determine the exact upper volume bound for such simplices and characterize the volume-maximizing simplices. We also determine the sharp upper bound on the coefficient of asymmetry of an integral polytope with a single interior integral point. This result confirms a conjecture of Hensley from 1983. Moreover, for an integral simplex with precisely one interior integral point, we give bounds on the volumes of its faces, the barycentric coordinates of the interior integral point and its number of integral points. Furthermore, we prove a bound on the lattice diameter of integral polytopes with a fixed number of interior integral points. The presented results have applications in toric geometry and in integer optimization. 相似文献
11.
Nathan Owen Ilten 《Journal of Pure and Applied Algebra》2009,213(6):1086-1096
In the case of two-dimensional cyclic quotient singularities, we classify all one-parameter toric deformations in terms of certain Minkowski decompositions introduced by Altmann [Minkowski sums and homogeneous deformations of toric varieties, Tohoku Math. J. (2) 47 (2) (1995) 151-184.]. In particular, we show how to induce each deformation from a versal family, describe exactly to which reduced versal base space components each such deformation maps, describe the singularities in the general fibers, and construct the corresponding partial simultaneous resolutions. 相似文献
12.
Sam Payne 《Mathematische Zeitschrift》2006,253(2):421-431
We show that the dual of the cone of divisors on a complete -factorial toric variety X whose stable base loci have dimension less than k is generated by curves on small modifications of X that move in families sweeping out the birational transforms of k-dimensional subvarieties of X. We give an example showing that it does not suffice to consider curves on X itself.
Supported by a Graduate Research Fellowship from the NSF 相似文献
14.
M. Brodmann 《Journal of Pure and Applied Algebra》2011,215(12):2859-184
Let X⊂Pr be a variety of almost minimal degree which is the projected image of a rational normal scroll from a point p outside of . In this paper we study the tangent spaces at singular points of X and the geometry of the embedding scrolls of X, i.e. the rational normal scrolls Y⊂Pr which contain X as a codimension one subvariety. 相似文献
15.
16.
Some years ago Caporaso and Harris have found a nice way to compute the numbers N(d, g) of complex plane curves of degree d and genus g through 3d + g − 1 general points with the help of relative Gromov-Witten invariants. Recently, Mikhalkin has found a way to reinterpret
the numbers N(d, g) in terms of tropical geometry and to compute them by counting certain lattice paths in integral polytopes. We relate these
two results by defining an analogue of the relative Gromov-Witten invariants and rederiving the Caporaso–Harris formula in
terms of both tropical geometry and lattice paths.
H. Markwig has been funded by the DFG grant Ga 636/2. 相似文献
17.
18.
We give the lower bound on the number of sharp shadow-boundaries of convexd-polytopes (or unbounded convex polytopal sets) withn facets. The polytopes (sets) attaining these bounds are characterized. Additionally, our results will be transferred to the dual theory.The research work of the first author was (partially) supported by Hungarian National Foundation for Scientific Research, grant no. 1812. 相似文献
19.
Birkett Huber Jörg Rambau Francisco Santos 《Journal of the European Mathematical Society》2000,2(2):179-198
In 1994, Sturmfels gave a polyhedral version of the Cayley Trick of elimination theory: he established an order-preserving
bijection between the posets of coherent mixed subdivisions of a Minkowski sum ?1+...+?
r
of point configurations and of coherent polyhedral subdivisions of the associated Cayley embedding ?(?1,...,?
r
). In this paper we extend this correspondence in a natural way to cover also non-coherent subdivisions. As an application, we show that the Cayley Trick combined with results of Santos on subdivisions of Lawrence
polytopes provides a new independent proof of the Bohne-Dress theorem on zonotopal tilings. This application uses a combinatorial
characterization of lifting subdivisions, also originally proved by Santos.
Received February 18, 1999 / final version received January 25, 2000?Published online May 22, 2000 相似文献
20.
We provide a detailed study of torsors over Laurent polynomial rings under the action of an algebraic group. As applications we obtained variations of Raghunathan’s results on torsors over affine space, isotriviality results for reductive group schemes and forms of algebras, and decomposition properties for Azumaya algebras. 相似文献