Chapter 1 PLANE CURVES
... The points of U 0 , those with x0 6= 0, are the points at finite distance. The points at infinity of P2 , those of the form (0, x1 , x2 ), are on the line at infinity ℓ0 , the locus {x0 = 0}, which is a projective line. This gives us a dichotomy: P2 is the union of the two sets U 0 and ℓ0 . We can a ...
... The points of U 0 , those with x0 6= 0, are the points at finite distance. The points at infinity of P2 , those of the form (0, x1 , x2 ), are on the line at infinity ℓ0 , the locus {x0 = 0}, which is a projective line. This gives us a dichotomy: P2 is the union of the two sets U 0 and ℓ0 . We can a ...
Invariants and Algebraic Quotients
... Proof. We will constantly use the fact that every G-module non-zero Gmodule contains a simple submodule. (i) ⇒ (ii): This is obvious from the definition. (ii) ⇒ (iii): If W ⊆ V is a strict submodule, then there is a simple submodule U ⊆ V not contained in W . It follows that U ∩ W = {0}, and so W + ...
... Proof. We will constantly use the fact that every G-module non-zero Gmodule contains a simple submodule. (i) ⇒ (ii): This is obvious from the definition. (ii) ⇒ (iii): If W ⊆ V is a strict submodule, then there is a simple submodule U ⊆ V not contained in W . It follows that U ∩ W = {0}, and so W + ...
Lines on Projective Hypersurfaces
... Since Σkp is the intersection of k hypersurfaces in Pn , its expected dimension is n − k. The next theorem says that generally the expected dimension is obtained unless the whole cone lies in X. Theorem 3.1 (Landsberg [10]). Let X ⊂ Pn be a hypersurface, and let p be a general point of X. Any irredu ...
... Since Σkp is the intersection of k hypersurfaces in Pn , its expected dimension is n − k. The next theorem says that generally the expected dimension is obtained unless the whole cone lies in X. Theorem 3.1 (Landsberg [10]). Let X ⊂ Pn be a hypersurface, and let p be a general point of X. Any irredu ...
Classification of complex projective towers up to dimension 8 and
... rigidity indeed fails to hold for CP -towers, and if so, exactly in what dimension, does it fail? In this paper, we answer these questions by complete classification of CP -towers up to dimension 6, and some special 2-stage CP -towers of dimension 8. We now describe our classification results. Note th ...
... rigidity indeed fails to hold for CP -towers, and if so, exactly in what dimension, does it fail? In this paper, we answer these questions by complete classification of CP -towers up to dimension 6, and some special 2-stage CP -towers of dimension 8. We now describe our classification results. Note th ...
PROJECTIVE MODULES AND VECTOR BUNDLES The basic
... modules are naturally identified with matrices over R. The canonical example of a based free module is Rn with the usual basis; it consists of n-tuples of elements of R, or “column vectors” of length n. Unfortunately, there are rings for which Rn ∼ = Rn+t , t 6= 0. We make the following definition t ...
... modules are naturally identified with matrices over R. The canonical example of a based free module is Rn with the usual basis; it consists of n-tuples of elements of R, or “column vectors” of length n. Unfortunately, there are rings for which Rn ∼ = Rn+t , t 6= 0. We make the following definition t ...
MA3A6 Algebraic Number Theory
... Proof. Firstly, let’s prove the “if” part. It suffices to show that if L | K is a finite extension and α ∈ L, then α is algebraic. Suppose [ L : K ] = d < ∞. Then the powers 1, α, α2 , . . . , αd are d + 1 elements of a d-dimensional vector space over K, so they must be linearly dependent: that is, ...
... Proof. Firstly, let’s prove the “if” part. It suffices to show that if L | K is a finite extension and α ∈ L, then α is algebraic. Suppose [ L : K ] = d < ∞. Then the powers 1, α, α2 , . . . , αd are d + 1 elements of a d-dimensional vector space over K, so they must be linearly dependent: that is, ...
Differential Calculus of Several Variables
... point (1, 2, 3, 4, 5, 6, 7, 8, 9, 10) is a point in 10-space. We will typically denote an arbitrary point in Euclidean n-space by (x1 , x2 , . . . , xn ). One-dimensional space is usually denoted by R rather than R1 , and a typical point such as (4) is just written as 4. In other words, one-dimensio ...
... point (1, 2, 3, 4, 5, 6, 7, 8, 9, 10) is a point in 10-space. We will typically denote an arbitrary point in Euclidean n-space by (x1 , x2 , . . . , xn ). One-dimensional space is usually denoted by R rather than R1 , and a typical point such as (4) is just written as 4. In other words, one-dimensio ...
MATH 436 Notes: Finitely generated Abelian groups.
... If |X| < ∞ then | ⊕x∈X Z/2Z| = 2|X| . Thus | ⊕y∈Y Z/2Z| is also finite and so |Y | < ∞ and 2|Y | = 2|X| which yields |X| = |Y | as desired. So it only remains to consider the case |X| = |Y | = ∞. Let Pf inite (X) = {S ⊂ X||S| < ∞} be the set of finite subsets of X. Then there is a bijection Θ : Pf i ...
... If |X| < ∞ then | ⊕x∈X Z/2Z| = 2|X| . Thus | ⊕y∈Y Z/2Z| is also finite and so |Y | < ∞ and 2|Y | = 2|X| which yields |X| = |Y | as desired. So it only remains to consider the case |X| = |Y | = ∞. Let Pf inite (X) = {S ⊂ X||S| < ∞} be the set of finite subsets of X. Then there is a bijection Θ : Pf i ...
Linear Differential Equations
... Remark. We luck out that these are all the solutions; there are no other solutions. This is not proven here; take my word for it. Remark. Among the many solutions x(t) = Ae−4t + Be−7t , ∀A and B , the two we found first, x1 = e−4t and x2 = e−7t , are indeed much simpler than the bulk of the others. ...
... Remark. We luck out that these are all the solutions; there are no other solutions. This is not proven here; take my word for it. Remark. Among the many solutions x(t) = Ae−4t + Be−7t , ∀A and B , the two we found first, x1 = e−4t and x2 = e−7t , are indeed much simpler than the bulk of the others. ...
Standard Monomial Theory and applications
... roots and characters, there exists an additional important combinatorial tool: the Young tableaux. To construct objects like the tableaux in a more general setting, consider the weight lattice X of a complex semisimple Lie algebra (or, more generallly, symmetrizable Kac-Moody algebra) g, and denote ...
... roots and characters, there exists an additional important combinatorial tool: the Young tableaux. To construct objects like the tableaux in a more general setting, consider the weight lattice X of a complex semisimple Lie algebra (or, more generallly, symmetrizable Kac-Moody algebra) g, and denote ...
Undergraduate algebra
... What are the symmetries of the rectangle? In any symmetry, a short side has to go to a short side, and a long side to a long side. Once a decision is made which side goes where, there is precisely one symmetry that does it (these facts are clear intuitively, and are easy to prove if the precise defi ...
... What are the symmetries of the rectangle? In any symmetry, a short side has to go to a short side, and a long side to a long side. Once a decision is made which side goes where, there is precisely one symmetry that does it (these facts are clear intuitively, and are easy to prove if the precise defi ...
Homology - Nom de domaine gipsa
... Morally, the homology groups count the number of holes in each dimension. But the situation is more subtle due to the torsion that may appear in the homology groups. We illustrate this on a few examples. In order to compute the homology of a surface, the first step could be to find a triangulation o ...
... Morally, the homology groups count the number of holes in each dimension. But the situation is more subtle due to the torsion that may appear in the homology groups. We illustrate this on a few examples. In order to compute the homology of a surface, the first step could be to find a triangulation o ...
Basis (linear algebra)
Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.A set of vectors in a vector space V is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set. In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space V, every element of V can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.