N be a rational map, where f 0 ,. .. , f N are homogeneous polynomials of degree d with no common factors. Then φ is defined at all points not in the indeterminacy locus Z(φ) = { P ∈ P N : f 0 (P) = · · · = f N (P) = 0 }. The map φ is said to be dominant if the image φ (P N Z(φ)) is Zariski dense, i.e., does not lie in a proper algebraic subset of P N. Alternatively, the map φ is dominant if th...

We prove that the Seidel morphism of (M × M , ω ⊕ ω) is naturally related to the Seidel morphisms of (M,ω) and (M , ω), when these manifolds are monotone. We deduce a condition for loops of Hamiltonian diffeomorphisms of the product to be homotopically non trivial. This result was inspired by and extends results obtained by Pedroza [P]. All the symplectic manifolds we consider in this note are ...

Theorem A. Let f : X → S be a flat and projective morphism of varieties over C with n = dim f . Let L0, . . . , Ln be C -hermitian line bundles on X. Then, the metric of Deligne’s pairing 〈L0, . . . , Ln〉(X/S) is continuous. Here we note that the C of hermitian line bundles in this note is slightly stronger than the sense of [11] and [8]. For simplicity, we explain it in terms of the C of funct...

Let X be a smooth complex projective variety with nef ⋀2TX and dim⁡X≥3. We prove that, up to finite étale cover X˜→X, the Albanese map X˜→Alb(X˜) is locally trivial fibration whose fibers are isomorphic Fano F ⋀2TF. As bi-product, we see that TX or variety. Moreover study contraction of KX-negative extremal ray φ:X→Y. In particular, blow-up space at point if φ birational type. also morphism fib...

Exercise 10.3.2. Let R be a commutative ring with identity. For all positive integers n and m, R ∼= R if and only if n = m. Proof. Let φ : R → R be an isomorphism of R-modules and let I E R be a maximal ideal. Then the map φ̄ : R → R/IR given by φ̄(α) = φ(α) is a morphism of R-modules. Moreover ker φ̄ = {α ∈ R | φ̄(α) = 0} = {α ∈ R | φ(α) ∈ IR} = φ−1(IRm) = IR. Therefore by the first isomorphism th...

For a site S (with enough points), we construct a topological space X(S) and a full embedding φ of the category of sheaves on S into those on X(S) (i.e., a morphism of toposes φ: Sh(X(S)) → Sh(S)). The embedding will be shown to induce a full embedding of derived categories, hence isomorphisms H(S, A) = H∗(X(S), φ A) for any abelian sheaf A on S. As a particular case, this will give for any sch...

An internal factor of a word x is a word v such that x = uvw for some nonempty words u, w. The kernel of a set X of words is the set of words of X which are internal factors of words of X. Let φ be the syntactic morphism of the submonoid X∗ generated by X. We prove that if X is a code with empty kernel, the groups contained in the image by φ of the complement of the set of internal factors of t...

We strengthen a result of Michiel Kosters by proving the following theorems: (*) Let φ: W → V be a finite surjective morphism of algebraic varieties over an ample field K. Suppose V has a simple K-rational point a such that a / ∈ φ(W (Kins)). Then, card(V (K)rφ(W (K)) = card(K). (**) Let K be an infinite field of positive characteristic and let f ∈ K[X] be a nonconstant monic polynomial. Suppos...

The Tango bundle T over P is proved to be the pull-back of the twisted Cayley bundle C(1) via a map f : P → Q5 existing only in characteristic 2. The Frobenius morphism φ factorizes via such f . Using f the cohomology of T is computed in terms of S ⊗C, φ∗(C), Sym(C) and C, while these are computed by applying Borel-Bott-Weil theorem. By machine-aided computation the mimimal resolutions of C and...

Let M be a complete Riemannian manifold and N a complete noncompact Riemannian manifold. Let φ : M → N be a surjective harmonic morphism. We prove that if N admits a subharmonic function with finite Dirichlet integral which is not harmonic, and φ has finite energy, then φ is a constant map. Similarly, if f is a subharmonic function on N which is not harmonic and such that |df | is bounded, and ...