Explore: Abelian Varieties And Schemes

defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for research

In algebraic geometry, a semistable abelian variety is an abelian variety defined over a global or local field, which is characterized by how it reduces

in the duality theory of abelian varieties. It was constructed by Grothendieck (1962), and also described by Mumford (1966) and Kleiman (2005). In the cases

higher-dimensional abelian varieties, so the concept of dual becomes more interesting in higher dimensions. Let A be an abelian variety over a field k. We

group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they

the duality theory of abelian varieties in characteristic p (theory of Pierre Cartier). The Galois cohomology of this group scheme is a way of expressing

Albanese variety of a smooth projective algebraic variety V {\displaystyle V} is an abelian variety Alb ⁡ ( V ) {\displaystyle \operatorname {Alb} (V)}

multiplicities (the equations x = 0 and x2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative

higher-dimensional projective varieties naturally leads to the construction of moduli of projective varieties. Hilbert schemes parametrize closed subschemes

principally polarized complex abelian varieties of dimension g {\displaystyle g} (a principal polarization identifies an abelian variety with its dual). The theory