Curves

Definitions

What is an algebraic curve?
What is the genus of a curve?
Special types of varieties:
- What is a Fano variety?
- What is a del Pezzo surface?
- What is a ruled variety?
- What is a uniruled variety?
- What is an elliptic curve?
- What is a hyperelliptic curve?
- What is an abelian variety?
- What is the Jacobian of a curve?
- What is the Jacobian variety?
- What is a Brill-Noether curve?
- What is a Severi-Brauer variety?
What is the special divisor?
What is the ramification divisor?
What is the Frobenius map?
What is the j invariant?
What is an elliptic function?
What is the Hasse invariant?
What is the canonical embedding?
What is the anticanonical embedding?
- The geometric genus?
- The arithmetic genus?
- The gonality?
- The degree of a morphism $C\to {\mathbb{P}}^r$?
What is the dual curve?
What is the resultant?
What is the Kodaira dimension?
What is the Todd genus?
What is a minimal model of a smooth projective variety?
What is an elliptic fibration?
Types of surfaces:
- An Enriques surface?
- A K3 surface?
- An elliptic surface?
- A hyperelliptic surface?
- An abelian surface?
- What is the Hirzebruch surface?
  - Examples:
What is a log resolution?
What is a rational curve?
What is a stable map?
What is a reducible curve?
What is devissage?
What is a crepant map?
What is a complete intersection?
What are ADE singularities?
What is the Fermat quartic?

What is Castelnuovo’s theorem?
What is the Riemann-Roch theorem for curves?
What is the Riemann-Hurwitz theorem for curves?
- $2-2 g_{1}=2\left(2-2 g_{0}\right)-\displaystyle\sum_{s \in X^{\mathrm{ram}}}\left(e_{s}-1\right)$
What is Hurwitz’s theorem?
What is the classification of curves in ${\mathbb{P}}^3$?
What is the Riemann-Hurwitz formula?
What is the genus formula for plane curves?
What is the Brill-Noether theorem?
Why is every genus 2 curve hyperellptic? Why is this not even generically true for $g\geq 3$?
How can you view a curve as a ramified cover of ${\mathbb{P}}^1_{/ {k}} $?
What is the cohomological criterion for ampleness?
Why is Zariski’s theorem true?
What is the Hodge index theorem?
What is Mori’s theorem?
What is Zariski’s main theorem?
What is the minimal model program for surfaces?

Show that if $C$ is a smooth proper curve of genus $g(C)\geq 2$, then ${\sharp}\mathop{\mathrm{Aut}}(G) < \infty$.
Show that given $\pi: X \rightarrow Y$, $Y$ is integrally closed in $K(X)$ over $K(Y)$ if and only if ${\mathcal{O}}_{Y} \rightarrow \pi_{*} {\mathcal{O}}_{X}$ is an isomorphism (i.e., $\pi$ is ${\mathcal{O}}$-connected).
Use Tsen’s theorem to show that given a flat family $X\to Y$ with $Y$ a curve where the fibers smooth curves of genus 0, this determines a Zariski ${\mathbb{P}}^1{\hbox{-}}$bundle iff there exists a relative degree 1 line bundle on $X$ over $Y$.