Angel "Java" Lopez en Blog

Publicado el 21 de Enero, 2018, 14:24

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Hace tiempo que no escribo del tema, es tiempo de retomar esta serie. Hay mucho para leer, explorar y estudiar sobre el tema. Mas que revisar cada libro, voy a comentar salteado distintas partes de libros. En estos días estuve leyendo a David Mumford (por ejemplo, su The Red Book of Varieties and Schemes). El desarrollo de la geometría algebraica ha sido notable durante el siglo XX, y justo en estos días encuentro una cita de Mumford, que me resulta importante para entender todo lo que fue pasando en el siglo pasado. Leo en su Algebraic geometry I, Complex Projective Varieties-Springer (1976), casi al comienzo:

... In the 20th century, algebraic geometry has gone through at least 3 distinct phases. In the period 1900-1930, largely under the leadership of the 3 Italians, Castelnuovo, Enriques and Severi, the subject grew immensely. In particular, what the late 19th century had done for curves, this period did for surfaces: a deep and systematic theory of surfaces was created. Moreover, the links between the "synthetic" or purely "algebro-geometric" techniques for studying surfaces, and the topological and analytic techniques were thoroughly explored. However the very diversity of tools available and the richness of the intuitively appealing geometric picture that was built up, led this school into short-cutting the fine details of all proofs and ignoring at times the time-consuming analysis of special cases (e.g., possibly degenerate configurations in a construction). This is the traditional difficulty of geometry, from High School Euclidean geometry on up. In the period 1930-1960, under the leadership of Zariski, Weil, and (towards the end) Grothendieck, an immense program was launched to introduce systematically the tools of commutative algebra into algebraic geometry and to find a common language in which to talk, for instance, of projective varieties over characteristic p fields as well as over the complex numbers. In fact, the goal, which really goes back to Kronecker, was to create a "geometry" incorporating at least formally arithmetic as well as projective geometry. Several ways of achieving this were proposed, but after a somewhat chaotic period in which communication was difficult, it seems fair to say that Grothendieck's "schemes" have become generally accepted as providing the most satisfactory foundations. In the present period 1960 on, algebraic geometry is growing rapidly in many directions at once: to a deeper understanding of geometry in dimensions higher than 2, especially their singularities, and the theory of cycles on them; to uncovering the astonishing connections between the topology of varieties and their Diophantine properties (their rational points over" finite fields and number fields); and to the theory of moduli, i.e., the parameters describing continuous families of varieties.


Escribe esto en 1975, pero ya se veía entonces esa división en etapas sobre el desarrollo de la geometría algebraica, en el siglo XX. Escribiendo en esa época, se le escapa cómo contribuyó este tema a la resolución del famoso último teorema de Fermat: hay igual que reconocer que ése es apenas uno de los temas en los que ha colaborado este "revival" desde la segunda etapa de arriba. Muchos de los desarrollos actuales, no hubieran sido posibles si no se hubiera extendido todo con las nuevas ideas, especialmente desde Grothrendieck.

Para entender la motivación y la importancia de la introducción de los schemes, ver

Why Schemes?

También ver

Basic Modern Algebraic Geometry
https://www.irif.fr/~mellies/mpri/mpri-ens/biblio/Audun-Holme-Basic-Modern-Algebraic-Geometry.pdf
Introduction to Grothendieck"s Theory of Schemes

Donde también tenemos una introducción a categorías.

Para completar la cita de arriba, leer

https://en.wikipedia.org/wiki/Algebraic_geometry#20th_century

Donde encuentro:

An important class of varieties, not easily understood directly from their defining equations, are the abelian varieties, which are the projective varieties whose points form an abelian group. The prototypical examples are the elliptic curves, which have a rich theory. They were instrumental in the proof of Fermat's last theorem and are also used in elliptic curve cryptography.

El desarrollo de las curvas elípticas ha sido notable, y como cita Wikipedia, han encontrado su utilización en la criptografía. Hoy, los famosos Bitcoins, tiene curva elíptica en el fondo de su seguridad. El que los puntos de una curva formen un grupo, es notable y muy interesante. El primer ejemplo lo tenemos en los puntos de una circunferencia. Y cuando los puntos son racionales, su producto por el grupo, es racional (corresponde a la suma de ángulos; en curvas elípticas es menos trivial).

Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
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