Algebraic geometry II - schemes and sheaves

1) Cohomological characterization of ampleness.

2) Every quasi-compact scheme has a closed point.

The rest is not part of exam syllabus.

3) Higer derived pushforwards of a coherent sheaf along a proper morphism is coherent (no proof).

4) Euler characteristic.

5) Riemann-Roch on curves which are proper over an algebraically closed field. 

6) Serre duality (no proof)

7) Consequences of Riemann-Roch.

8) A morphism separating points and tangent vectors is a closed embedding.

9) Any regular, integral curve which is proper over an algebraically closed field is projective.