---
title: Flashcards Birational
aliases: 
flashcard: Research::April_2024
created: 2023-03-26T11:58
updated: 2024-05-10T12:27
---


# Flashcards Birational

- [ ] What is the Nef cone?
- [ ] What is the ample cone?
- [x] What is Castelnuovo's contractibility theorem? ✅ 2024-04-19
	- [ ] If $E\subseteq S$ a smooth surface with $E^2=-1$ then there exists a birational contraction $S\to S'$ contracting only $E$ where $S'$ is again smooth.
- [x] Why is $E^2 = -1$ in a blowup? ✅ 2024-04-19
	- [ ] Take a tubular neighbourhood of $E$, perturbing $E$  to $E'$ introduces a pole like $1/z$, scaling by $\abs{z}$ yields $\bar{z}$, so $E'$ becomes antiholomorphic and the orientation is reversed.
	- [ ] ![](2024-04-19.png)
- [x] What is the Nakai-Moishezon criterion? ✅ 2024-04-21
	- [ ] Relates ampleness with positivity.
	- [ ] $D\in \Amp(X) \iff \forall V\leq X$ irreducible subvarieties, $D^{n-k}\cdot V > 0$ where $\codim_X(V) = k$.
	- [ ] Thus ampleness only depends on the numerical class of $D$.
- [ ] What is Kleiman's criterion?
	- [ ] Ampleness can be tested on curves.
- [ ] What is a Mori dream space?
- [ ] What is a klt singularity?
- [ ] What is the Lefschetz hyperplane theorem?
- [ ] What is Kawamata-Viehweg vanishing?
- [ ] What is a multiplier ideal?
- [ ] What is the McKay correspondence?
- [ ] What is the Fujita conjecture?
- [ ] What is the abundance conjecture?
- [x] When is $\mcl \in \Pic(C)$ nef? Ample? Big? ✅ 2024-04-19
	- [ ] Nef: $\deg \mcl \geq 0$
	- [ ] Ample: $\deg \mcl > 0$
	- [ ] Big = Ample
- [ ] Define $\NS(X)$
- [ ] Define $\Pic(X)$ in terms of divisors and equivalence.
- [x] How are the ample and nef cones related? ✅ 2024-04-19
	- [ ] $\Amp(X) = \Nef(X)^\circ$.
- [x] Describe the transforms of lines in $\Bl_p \PP^2$ which (a) pass through $p$ and (b) do not pass through $p$. ✅ 2024-04-20
	- [ ] If $L$ passes through $p$, then $\pi^* L = L - E$
	- [ ] If $L$ does not, then $\pi^* L = L$.
- [ ] What is a Mori fibre space?
- [x] How does $\rank \NS(X)$ change under contractions of curves? ✅ 2024-04-21
	- [ ] Strictly decreases
- [x] What is a minimal surface? ✅ 2024-04-21
	- [ ] $K_X \in \Nef(X)$.
- [x] What is a stable curve? ✅ 2024-04-21
	- [ ] $\Aut(X) < \infty$ or $K_X \in \Amp(X)$ (e.g. when $g\geq 2$).
- [x] What is $\Amp(X)\dual$? ✅ 2024-04-21
	- [ ] The cone of curves.
- [ ] What is Mori's cone theorem?
- [ ] What is a pseudo-effective divisor?
- [ ] What is a semiample divisor?
- [ ] What is a Gorenstein variety?
- [x] How are terminal singularities analyzed? ✅ 2024-04-21
	- [ ] Take a resolution of singularities $Y\mapsvia{f} X$ and write $K_Y = f^* K_X + \sum m_i E_i$.
	- [ ] Terminal: $m_i > 0$
	- [ ] Canonical: $m_i \geq 0$
	- [ ] Log terminal: $m_i > -1$.
- [ ] What is a flop?
	- [ ] Contract a curve $C$ satisfying $K_X\cdot C = 0$.
- [ ] What's the best way to prove that a variety is projective?
	- [ ] Find $D\in \Amp(X)$.
- [ ] What is asymptotic Riemann-Roch?
	- [ ] $\chi(\OO_X(mD)) = {D^n \over n!}m^n + \bigo(m^{n-1})$.