LECTURES ON ALGEBRAIC VARIETIES OVER F1 Contents
... Thirty five years later, Smirnov [7], and then Kapranov and Manin, wrote about F1 , viewed as the missing ground field over which number rings are defined. Since then several people studied F1 and tried to define algebraic geometry over it. Today, there are at least seven different definitions of su ...
... Thirty five years later, Smirnov [7], and then Kapranov and Manin, wrote about F1 , viewed as the missing ground field over which number rings are defined. Since then several people studied F1 and tried to define algebraic geometry over it. Today, there are at least seven different definitions of su ...
Axiomatising the modal logic of affine planes
... A2. there is a unique line through any point parallel to any given line ∀x ∈ P ∀l ∈ L ∃ ! m ∈ L(x E m ∧ m || l) A3. there exist three non-collinear points ∃x0 x1 x2 ∈ P ¬∃l ∈ L(x0 E l ∧ x1 E l ∧ x2 E l) (equivalently: L 6= ∅, and for any l ∈ L, there is a point not on l) ...
... A2. there is a unique line through any point parallel to any given line ∀x ∈ P ∀l ∈ L ∃ ! m ∈ L(x E m ∧ m || l) A3. there exist three non-collinear points ∃x0 x1 x2 ∈ P ¬∃l ∈ L(x0 E l ∧ x1 E l ∧ x2 E l) (equivalently: L 6= ∅, and for any l ∈ L, there is a point not on l) ...
4.1 Introduction to Linear Spaces
... What are all of the vector subspaces of R3? A) The zero vector B) Any line passing through the origin C) Any plane containing the origin. D) All of R3 ...
... What are all of the vector subspaces of R3? A) The zero vector B) Any line passing through the origin C) Any plane containing the origin. D) All of R3 ...
PDF
... integral and quasi-projective) over a field K with characteristic zero. We regard V as a topological space with the usual Zariski topology. 1. A subset A ⊂ V (K) is said to be of type C1 if there is a closed subset W ⊂ V , with W 6= V , such that A ⊂ W (K). In other words, A is not dense in V (with ...
... integral and quasi-projective) over a field K with characteristic zero. We regard V as a topological space with the usual Zariski topology. 1. A subset A ⊂ V (K) is said to be of type C1 if there is a closed subset W ⊂ V , with W 6= V , such that A ⊂ W (K). In other words, A is not dense in V (with ...
Solutions Sheet 7
... 5. Let X be a locally noetherian scheme. Prove that the set of irreducible components of X is locally finite, i.e. that every point of X has an open neighborhood which meets only finitely many irreducible components of X. Solution: By definition the irreducible components of a topological space are ...
... 5. Let X be a locally noetherian scheme. Prove that the set of irreducible components of X is locally finite, i.e. that every point of X has an open neighborhood which meets only finitely many irreducible components of X. Solution: By definition the irreducible components of a topological space are ...
Affine space
In mathematics, an affine space is a geometric structure that generalizes certain properties of parallel lines in Euclidean space. In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead displacement vectors between two points of the space. Thus it makes sense to subtract two points of the space, giving a vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a vector to a point of an affine space, resulting in a new point displaced from the starting point by that vector. The simplest example of an affine space is a linear subspace of a vector space that has been translated away from the origin. In finite dimensions, such an affine subspace corresponds to the solution set of an inhomogeneous linear system. The displacement vectors for that affine space live in the solution set of the corresponding homogeneous linear system, which is a linear subspace. Linear subspaces, in contrast, always contain the origin of the vector space.