2022-07-07

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2022-07-07

Current reading: Voison on the Luroth problem

  • What does it mean for a variety to be unirational?
  • What is a dominant rational map?
  • What is a blowup along a smooth center?
  • When does unirationality imply rationality?
  • What methods were used to determine solutions to the Luroth problem in dimensions \(d\geq 3\)?
  • What is a very general hypersurface?
  • Give examples of cubic hypersurfaces.
  • State a form of GAGA.
  • What is a rationally connected variety?
  • What is the Brauer group?
  • What is the Artin-Mumford invariant?
  • Describe \({\mathcal{O}}_{{\mathbf{P}}^n}(m)\) in equations.
  • What is Abel theorem?
  • What is Bloch’s famouse conjecture?
    • \(H^0(X; \Omega_{X_{/ {k}} }^{\geq 1}) = 1 \implies {\operatorname{CH}}_0(X) = 1\)
  • What is map identifying \({\operatorname{CH}}^1(X) { \, \xrightarrow{\sim}\, }\operatorname{Pic}(X)\)?
  • Describe the Hodge decomposition and the Hodge diamond.
  • What is an integral Hodge class?
  • What is the Lefschetz (1, 1) theorem?
  • What is the cycle class map? What is its image?
  • What is unramified cohomology?
  • What is the intermediate Jacobian?
  • What is the Lefschetz decomposition on homology?
  • What is the Abel-Jacobi map?
  • What is the \({\partial}{ \mkern 1.5mu\overline{\mkern-1.5mu{\partial}\mkern-1.5mu}\mkern 1.5mu}\) lemma?
  • What is the theta divisor?
  • What is a unimodular intersection pairing?
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