# Algebraic Geometry

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**Algebraic Geometry**In this paper we first generalize to the case of partial flags a result proved both by Spaltenstein and by Steinberg that relates the relative position of two complete flags and the irreducible components of the flag variety in which they... more

We consider a certain class of Schubert varieties of the affine Grassmannian of type A. By embedding a Schubert variety into a finite-dimensional Grassmannian, we construct an explicit basis of sections of the basic line bundle by... more

Solution algebras can be associated to any connection over a smooth affine variety. It turns out that he spectrum of a solution algebra is an algebraic fiber space over the base variety, with quasi-homogeneous fiber. We discuss the... more

We show that a variety of monodromy phenomena arising in geometric topology and algebraic geometry are most conveniently described in terms of homomorphisms from a(n augmented) knot quandle associated with the base to a suitable... more

In this essay, we review the Nikiforov-Uvarov method which is used to solve Schro¨dinger equation. We shed light on the algorithm from the viewpoint of algebraic geometry so that we shall the ideas of the latter (such as the resolution of... more

The Alexander-Hirschowitz theorem says that a general collection of $k$ double points in ${\bf P}^n$ imposes independent conditions on homogeneous polynomials of degree $d$ with a well known list of exceptions. Alexander and Hirschowitz... more

A theorem of Macaulay on colons of ideals in polynomial rings is proved for homogeneous Gorenstein algebras.

In this paper we give lower bounds for the minimum distance of evaluation codes constructed from complete intersections in toric varieties. This generalizes the results of Gold-Little-Schenck and Ballico-Fontanari who considered... more

This paper is the widely extended version of the publication, appeared in Proceedings of ISSAC'2009 conference \citep*{ALM09}. We discuss more details on proofs, present new algorithms and examples. We present a general algorithm for... more

We present in this paper a differential version of Mirzakhani's recursion relation for the Weil-Petersson volumes of the moduli spaces of bordered Riemann surfaces. We discover that the differential relation, which is equivalent to the... more

Abstract. Let k be a field and let F⊂ k [x1,..., xn] be a finite set of monomials whose exponents lie on a positive hyperplane. We give necessary conditions for the normality of both the Rees algebra R [Ft] and the subring k [F]. If the... more

There is the possibility of generalization of Calabi-Yau spaces by an abstact point of view.

Phylogenetic mixture models are statistical models of character evolution allowing for heterogeneity. Each of the classes in some unknown partition of the characters may evolve by different processes, or even along different trees. The... more

A century and a half ago, a revolution in human thought began that has gone largely unrecognized by modern scholars: A system of non-Euclidean geometries was developed that literally changed the way that we view our world. At first, some... more

We piece together ingredients, which are well known and documented in the literature, into a new proof of the existence of semistable 3-fold flips

The paper is a second step in the study of $\overline{M}_{0,n}$ started in arXiv:1006.0987 [math.AG]. We study fiber type morphisms of this moduli space via Kapranov's beautiful description. Our final goal is to understand if any... more

In this paper, we discuss the theory of the Siegel modular variety in the aspects of arithmetic and geometry. This article covers the theory of Siegel modular forms, the Hecke theory, a lifting of elliptic cusp forms, geometric properties... more

People these days know the Universe as a Whole, because not knowing the edge between This and That. Its secret is the secret of a forgotten body, the seventh in the series of multifaceted as Life, Seven. We know six of it now. But the... more

In the first part of this paper we study scrollers and linearly joined varieties. A particular class of varieties, of important interest in classical Geometry are Cohen--Macaulay varieties of minimal degree. They appear naturally studying... more

Lógica, Números, Funciones, Conjuntos, propiedades de las operaciones entre conjuntos, cuantificadores, números reales, expresiones algebraicas, valor absoluto, Ecuaciones, Inecuaciones, Técnicas de conteo, Teorema del Binomio,... more

These are my original research of elementary complex numbers applied on the triangle. The vertices of the triangle are given as the initial elements with which in the complex plane are defined sides, area, center of gravity, orthocenter... more

Proporción y sucesion de Fibonacci en la espiral áurea.

The theory of Lambda-rings, in the sense of Grothendieck's Riemann-Roch theory, is an enrichment of the theory of commutative rings. In the same way, we can enrich usual algebraic geometry over the ring Z of integers to produce... more

These notes are based on a lecture course by L. Chekhov held at the University of Manchester in May 2006 and February-March 2007. They are divulgative in character, and instead of containing rigorous mathematical proofs, they illustrate... more