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The first key result is the cone theorem of Shigefumi Mori, describing the structure of the cone of curves of . Briefly, the theorem shows that starting with X {\displaystyle X} , one can inductively construct a sequence of varieties X i {\displaystyle X_{i}} , each of which is "closer" than the previous one to having K X i {\displaystyle K_{X ...
A variety X over an uncountable algebraically closed field k is uniruled if and only if there is a rational curve passing through every k-point of X. By contrast, there are varieties over the algebraic closure k of a finite field which are not uniruled but have a rational curve through every k -point.
Faltings's theorem is a result in arithmetic geometry, according to which a curve of genus greater than 1 over the field of rational numbers has only finitely many rational points. This was conjectured in 1922 by Louis Mordell , [ 1 ] and known as the Mordell conjecture until its 1983 proof by Gerd Faltings . [ 2 ]
This is an improvement of many classical results, beginning with the case of cubic surfaces (which are rational varieties over an algebraic closure). Other examples of varieties that are shown to be unirational are many cases of the moduli space of curves. [2]
Jacobians of curves are naturally equipped with a principal polarisation as soon as one picks an arbitrary rational base point on the curve, and the curve can be reconstructed from its polarised Jacobian when the genus is >. Not all principally polarised abelian varieties are Jacobians of curves; see the Schottky problem.
Gröbner basis computation is one of the main practical tools for solving systems of polynomial equations and computing the images of algebraic varieties under projections or rational maps. Gröbner basis computation can be seen as a multivariate, non-linear generalization of both Euclid's algorithm for computing polynomial greatest common ...
The case with an elliptic curve and the field of rational numbers is Mordell's theorem, answering a question apparently posed by Henri Poincaré around 1901; it was proved by Louis Mordell in 1922. It is a foundational theorem of Diophantine geometry and the arithmetic of abelian varieties .
The modularity of an elliptic curve E of conductor N can be expressed also by saying that there is a non-constant rational map defined over ℚ, from the modular curve X 0 (N) to E. In particular, the points of E can be parametrized by modular functions. For example, a modular parametrization of the curve y 2 − y = x 3 − x is given by [18]