User:Ilya (usurped)/AG

Algebraic Geometry for Dummies (with pictures and dialogues)

Understanding Curves

• classical conics • curve=equation over C • genus calculations • covering P¹ • fundamental groups • algebraic transformations • everything is plain • one-dimensional complex variety • morphisms to P¹ • algebraic forms • the sphere is round • Jac X • natural immersion • how many rational points? • Abel-Jacobi • la mer était étale... •

What's hidden behind 2x2=4?

• picture of Spec Z[i] • dialogue about division • finite fields • Frobenius morphism • Galois correspondence •

Schematic zoo

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

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 • 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

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

Derived, Differential, Dao or Looking Inside The Mirror

• coherent sheaves are complexes of bundles • why derived categories? • six operations • generators on Pⁿ • Fourier transformation • D-modules •

Motives, Mixed, Mystery or New Age Arithmetics

• cut and paste • counting F_p^n-rational points • L-functions • MHS • abelian motives • conjectures •

Grothendieck, Galois, Group or Dessins d'Enfants

• abelian vs. anabelian • simplest coverings • The Mapping Group and Other Adventures • they are arithmetical! • quest for absolute (Gal) • Galois-Teichmuller • Grothendieck-Kontsevich •