Abelian varieties Sample Clauses
Abelian varieties. In this chapter, we refer to ▇▇▇▇ [20] and ▇▇▇▇▇▇▇–Taniyama [40]. Let k be a field. In this thesis, we will use the following definitions. By a variety over k, we mean a geometrically integral, separated scheme of finite type over Spec(k). Curves, respectively surfaces, respectively threefolds are varieties are of dimension 1, respectively 2, respectively 3. We will always assume that curves, surfaces and threefolds are projective, smooth over k. By an abelian variety over k, we mean a complete irreducible group variety over k. It is known that abelian varieties are smooth, projective, and commutative. Let A and B be abelian varieties over k. A morphism λ of A to B is a morphism of varieties that respects the group structure. If A and B are of the same dimension and λ is surjective, then it is called an isogeny. If an isogeny λ : A B exists, then A and B are called isogenous. A non-zero abelian variety is said to be simple if it is not isogenous to a product of abelian varieties of lower dimensions. We denote by End(A) the ring of homomorphisms of A to itself over k and we put End0(A) = End(A) ⊗ Q.
1.3.1 Polarizations and the dual variety This section basically follows Lang [20, 3.4]. By a divisor on a variety, we always mean a Cartier divisor. We say that two divisors X1 and X2 on an abelian variety A over a field k are algebraically equivalent (X1 X2) if there is a connected algebraic set T , two points t1, t2 T and a divisor Z on A T such that Z t = Xi for i = 1, 2. The divisors X1 and X2 are linearly equivalent if there is a rational function f k(A)× such that X1 = X2 + (f ). For details, see Hartshorne [15].
Abelian varieties. We will give here a sketchy introduction to abelian varieties. For proofs and a more detailed theory we refer to ▇▇▇▇▇▇▇’s book [Mum08].
2.0.1 (Abelian variety). An abelian variety A over a field k is a projective variety with compatible group structure, that is, a projective variety over k, together with k-morphisms i∶ A → A, m∶ A × A → A, e∶ A → k, One can prove that the group law defined above is commutative. We will write the group operation additively. There is an analogue to abelian varieties over any base scheme. Definition 2.0.2 (Abelian scheme). An abelian scheme over a base scheme S of relative dimension g is a proper, smooth group scheme over S whose geometric fibers are connected and of dimension g. In other words an abelian scheme over S is a family of abelian varieties parametrized by S. An abelian scheme over a field k is an abelian variety as defined in Definition (2.0.1).