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2022-07-07

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?