(f) Explicitly verify that the the 5 global sections above generate the invertible sheaf OP2(2)|X(−C) at every point of X. Therefore there is a unique morphism φ : X → P such that φOP4(1) equals OP2(2)|X(−C) and the pullback of the homogeneous coordinates are the 5 sections above. (g) Also verify that y0, y1, y2, y3 are identically zero on the line L. Therefore φ contracts L to the point p = [0...

Abstract. Let φ : PNK → P N K be a morphism of degree d ≥ 2 defined over a field K that is algebraically closed field and complete with respect to a nonarchimedean absolute value. We prove that a modified Green function ĝφ associated to φ is Hölder continuous on P (K) and that the Fatou set F(φ) of φ is equal to the set of points at which ĝΦ is locally constant. Further, ĝφ vanishes precisely o...

Premorphisms are monotonic mappings between partially ordered monoids where the morphism condition φ(xy) = φ(x)φ(y) is relaxed into the condition φ(xy) ≤ φ(x)φ(y). Their use in place of morphisms has recently been advocated in situations where classical algebraic recognizability collapses. With languages of overlapping tiles, an extension of classical recognizability by morphisms, called quasi-...

In this paper, starting with a simply laced root system, we define a triangulated category which we call the m-cluster category, and we show that it encodes the combinatorics of the m-clusters of Fomin and Reading in a fashion similar to the way the cluster category of Buan, Marsh, Reineke, Reiten, and Todorov encodes the combinatorics of the clusters of Fomin and Zelevinsky. For Φ any root sys...

Definition 1.1. A group G is a set with a binary operation G × G → G, called multiplication, such that (1) ∀f, g, h ∈ G.(fg)h = f(gh) (2) ∃1 ∈ Gs.t.∀g ∈ G, 1g = g1 = g (3) ∀g ∈ G, ∃g−1 ∈ Gs.t. gg−1 = g−1g = 1 A morphism of groups φ : G → H is a function φ : G → H s.t. φ(g1g2) = φ(g1)φ(g2)∀g1, g2 ∈ G. Example 1.2. Z the group of integers, Z/nZ = the cyclic group of order n, Sym(X)the group of al...

Definition 1. Let S be a graded ring, set X = ProjS and letM a graded S-module. We define a sheaf of modulesM ̃ on X as follows. For each p ∈ ProjS we have the local ring S(p) and the S(p)module M(p) (GRM,Definition 4). Let Γ(U,M ̃) be the set of all functions s : U −→ ∐p∈U M(p) with s(p) ∈M(p) for each p, which are locally fractions. That is, for every p ∈ U there is an open neighborhood p ∈ V ⊆...

Let G = SpecA be a linearly reductive group, and wG ∈ A ∗ be G-invariant and wG(1) = 1. We establish the harmonic analysis on G and we compute wG when G = Sln, Gln, On, Sp2n, by geometric arguments and by means of the Fourier transform. Introduction An affine k-group G = SpecA is semisimple if and only if A splits in the form A = k ×B∗ as k-algebras, where the first projection π1 : A → k is the...

Let G be a finite group and φ : V → W an equivariant morphism of finite dimensional G-modules. We say that φ is faithful if G acts faithfully on φ(V ). The covariant dimension of G is the minimum of the dimension of φ(V ) taken over all faithful φ. In this paper we investigate covariant dimension and are able to determine it for abelian groups and to obtain estimates for the symmetric and alter...

The infinite Fibonacci sequence F, which is an extension of the classic Fibonacci sequence to the infinite alphabet N, is the fixed point of the morphism φ: (2i) 7→ (2i)(2i+ 1) and (2i+ 1) 7→ (2i+ 2) for all i ∈ N. In this paper, we study the growth order and digit sum of F, and give several decompositions of F using singular words.

In this paper we study the log structures on generalized semistable varieties, generalize the result by F. Kato and M. Olsson, and prove the canonicity of log structure when it can be expected. In out text we first give the definitions of local chart and weakly normal crossing morphism. Then we study the invariants of complete noetherian local ring coming from weakly normal crossing morphisms. ...