---
created: 2022-12-11T00:39
updated: 2023-12-28T09:19
tags:
  - study-guides
aliases:
  - curves and surfaces
---

# Curves and Surfaces Study Guide


- [ ]   What is the **Nullstellensatz**? How is it proved?
- [ ]   What is the **adjunction formula**?
- [ ]   What is the **Riemann-Roch theorem**?
    - [ ]   What is Riemann-Roch for curves?
    - [ ]   What is Riemann-Roch for surfaces?
- [ ]   What is the **Noether formula**?
- [ ]   What is **Serre duality**?
- [ ]   What is **Kodaira vanishing**?
- [ ]   What is the **Hilbert basis theorem**?
- [ ]   What is **Noether normalization**?
- [ ]   What is **Zariski's main theorem**?
- [ ]   What is the **Stein factorization** theorem?
- [ ]   What is the **Lefschetz hyperplane section** theorem?
- [ ]   What is the **moving lemma**?
- [ ]   What is the **Néron-Severi theorem**?
- [ ]   What is **Bertini's theorem**?
- [ ]   What is the **Cartan-Serre theorem**?
- [ ]   What is the **Nakai-Moishezon** criterion?
- [ ]   What is the **minimal model theorem for curves**?
- [ ]   What is the **minimal model program** for surfaces?
- [ ]   What is the **going down theorem**?
- [ ] What is Hurwitz's theorem?

- [ ] What is an **algebraic curve**?
- [ ] What is the **genus of a curve**?
- [ ] What is a [Fano](Fano) variety?
- [ ] What is a [del Pezzo](del%20Pezzo) surface?
- [ ] What is a **ruled** variety?
- [ ] What is a **uniruled** variety?
- [ ] What is an [elliptic curve](elliptic%20curve.md)?
- [ ] What is a [hyperelliptic](hyperelliptic) curve?
- [ ] What is an [abelian variety](abelian%20variety.md)?
- [ ] What is the [Jacobian](Jacobian.md) of a curve?
- [ ] What is the **Jacobian variety**?
- [ ] What is a [Brill-Noether curve](Brill-Noether%20curve)?
- [ ] What is a [Severi-Brauer variety](Severi-Brauer%20variety)?
- [ ] What is the [ramification divisor](ramification%20divisor.md)?
- [ ] What is the **Frobenius map**?
- [ ] What is the [j invariant](j%20invariant.md)?
- [ ] What is an [elliptic function](elliptic%20function)?
- [ ] What is the [Hasse invariant](Hasse%20invariant.md)?
- [ ] What is the [canonical embedding](canonical%20embedding)?
- [ ] What is the [anticanonical embedding](anticanonical%20embedding)?
- [ ] The [geometric genus](geometric%20genus)?
- [ ] The [arithmetic genus](arithmetic%20genus.md)?
- [ ] The [gonality](gonality)?
- [ ] The [degree](Unsorted/degree%20of%20a%20morphism.md) of a morphism $C\to {\mathbb{P}}^r$?
- [ ] What is the **dual curve**?
- [ ] What is the **resultant**?
- [ ] What is the [Kodaira dimension](Kodaira%20dimension.md)?
- [ ] What is the **Todd genus**?
- [ ] What is a [minimal model](minimal%20model) of a smooth projective variety?
- [ ] What is an [elliptic fibration](elliptic%20fibration)?
- [ ] An [Enriques surface](Enriques%20surface.md)?
- [ ] A [K3 surface](K3%20surface.md)?
- [ ] An [elliptic surface](elliptic%20surface)?
- [ ] A [hyperelliptic surface](hyperelliptic%20surface)?
- [ ] An **abelian surface**?
- [ ] What is the [Hirzebruch surface](Hirzebruch%20surface)?
- [ ] What is a [log resolution](log%20resolution.md)?
- [ ] What is a [rational curve](rational%20curve)?
- [ ] What is a [stable map](stable%20map.md)?
- [ ] What is a [reducible curve](reducible%20curve)?
- [ ] What is [devissage](Unsorted/devissage.md)?
- [ ] What is a [crepant](crepant) map?
- [ ] What is a [complete intersection](complete%20intersection)?
- [ ] What are [ADE singularities](ADE%20singularities)?
- [ ] What is the [Fermat quartic](Fermat%20quartic.md)?