User:Ilya (usurped)/AG

Algebraic Geometry for Dummies (with pictures and dialogues)

 * Understanding Curves
 * &bull; classical conics &bull; curve=equation over C &bull; genus calculations &bull; covering P1 &bull; fundamental groups &bull; algebraic transformations &bull; everything is plain &bull; one-dimensional complex variety &bull; morphisms to P1 &bull; algebraic forms &bull; the sphere is round &bull; Jac X &bull; natural immersion &bull; how many rational points? &bull; Abel-Jacobi &bull; la mer &eacute;tait &eacute;tale... &bull;


 * What's hidden behind 2x2=4?
 * &bull; picture of Spec Z[i] &bull; dialogue about division &bull; finite fields &bull; Frobenius morphism &bull; Galois correspondence &bull;


 * Schematic zoo
 * &bull; algebro-geometric dictionary &bull; always together: nilpotent schemes &bull; smoothness: the local ring &bull; are you normal? &bull; point over a point or visit to Galois &bull; algebraic numbers: the smallest schemes &bull; cross-breeding: base change &bull; Spec Z: knots without knots &bull; exotic species: non-algebraicity &bull; wild schemes: infinity &bull; why topoi? &bull;

Missing Parts
Here are some fairly standard topics that are covered in excellent notes (advised to be read by students independently during or after the course):
 * 1) Sheaf Cohomology: L'vovsky &bull; Hartschorne
 * 2) Elliptic Curves: J. Milne's home page
 * 3) Modular Forms: Stein notes on Ribet course

The Alphabet

 * Kummer, Kähler, Kodaira or The inner life of K3
 * &bull; orbifold E&times;E/Z2 (with pictures) &bull; resolution of singularities &bull; K=0: eternal curse or great gift? &bull; explicit equation: calculating K &bull; hypersections and their fundamental group &bull; sl2 action on Hodge cohomology &bull; Kodaira-Spencer: calculating the family dimension &bull; how's hard to be algebraic &bull; more elliptic bundles &bull; four-dimensional topology and integer forms (why signature of Calabi-Yau ? variety must be divisible by 8) &bull;


 * Derived, Differential, Dao or Looking Inside The Mirror
 * &bull; coherent sheaves are complexes of bundles &bull; why derived categories? &bull; six operations &bull; generators on Pn &bull; Fourier transformation &bull; D-modules &bull;


 * Motives, Mixed, Mystery or New Age Arithmetics
 * &bull; cut and paste &bull; counting F_p^n-rational points &bull; L-functions &bull; MHS &bull; abelian motives &bull; conjectures &bull;


 * Grothendieck, Galois, Group or Dessins d'Enfants
 * &bull; abelian vs. anabelian &bull; simplest coverings &bull; The Mapping Group and Other Adventures &bull; they are arithmetical! &bull; quest for absolute (Gal) &bull; Galois-Teichmuller &bull; Grothendieck-Kontsevich &bull;