Foreword These notes cover a short course on symplectic toric manifolds, delivered in six lectures at the summer school on Symplectic Geometry of Integrable Hamiltonian Systems, mostly for graduate students, held at the Centre de Recerca Matemàtica in Barcelona in July of 2001. The goal of this course is to provide a fast elementary introduction to toric manifolds (i.e., smooth toric varieties)...

Each infinitesimally faithful representation of a reductive complex connected algebraic group G induces a dominant morphism Φ from the group to its Lie algebra g by orthogonal projection in the endomorphism ring of the representation space. The map Φ identifies the field Q(G) of rational functions on G with an algebraic extension of the field Q(g) of rational functions on g. For the spin repres...

Coxeter-Petrie complexes naturally arise as thin diagram geometries whose rankthree residues contain all of the dual forms of a regular algebraic mapM. Corresponding to an algebraic map is its classical dual, which is obtained simply by interchanging the vertices and faces, as well as its Petrie dual, which comes about by replacing the faces by the so-called Petrie polygons. G. A. Jones and J. ...

In this paper a fundamental duality is established between algebraic cycles and algebraic cocycles on a smooth projective variety. The proof makes use of a new Chow moving lemma for families. If X is a smooth projective variety of dimension n, our duality map induces isomorphisms LH(X) → Ln−sH2n−k(X) for 2s ≤ k which carry over via natural transformations to the Poincaré duality isomorphism H(X...

A finite Blaschke product, restricted to the unit circle, is a smooth covering map. The maximum and minimum values of derivative this map reflect geometry product. We identify two classes extremal products: those that maximize difference between derivative, minimize it. Both turn out have same algebraic structure, being quotient hypergeometric polynomials.

This work describes a knowledge-guided clustering approach for mineral potential mapping (MPM), by which the optimum number of clusters is derived form a knowledge-driven methodology through a concentration-area (C-A) multifractal analysis. To implement the proposed approach, a case study at the North Narbaghi region in the Saveh, Markazi province of Iran, was investigated to discover porphyry ...

The classical (m, k)-Landen transform Fm,k is a self-map of the field of rational functions C(z) obtained by forming a weighted average of a rational function over twists by m’th roots of unity. Identifying the set of rational maps of degree d with an affine open subset of P, we prove that Fm,0 induces a dominant rational self-map Rd,m,0 of P 2d+1 of algebraic degree m, and for 1 ≤ k < m, the t...

Using only basic topological properties of real algebraic sets and regular morphisms we show that any injective regular self-mapping of a real algebraic set is surjective. Then we show that injective morphisms between germs of real algebraic sets define a partial order on the equivalence classes of these germs divided by continuous semi-algebraic homeomorphisms. We use this observation to deduc...

1.2. We refer to [LMB] for the basic background on stacks. Let us quickly recall: a stack over a scheme S is a sheaf of groupoids on the faithfully flat topology of S-schemes, where the presheaf requirement that the composition of the restriction maps is the restriction map of the composition is understood in the weak sense. Stacks form a 2-category with fiber products and there is a clear embe...

In this thesis we discuss a moduli space of projective curves with a map to a given projective space P. The functor CM parametrizes curves, that is, Cohen-Macaulay schemes of pure dimension 1, together with a finite map to P that is an isomorphism onto its image away from a finite set of closed points. We proof that CM is an algebraic space by contructing a scheme W and a representable, surject...