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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?