---
created: 2024-05-24T11:17
updated: 2024-05-28T20:09
flashcard: Review::Surfaces
---

# Misc

- [x] Give the formula for $\mcl \cdot \mcm \in \Pic(X)$ in terms of $\chi$. ✅ 2024-05-28
	- [ ] $\chi\left(\mathcal{O}_{X}\right)-\chi\left(\mathcal{L}^{-1}\right)-\chi\left(\mathcal{M}^{-1}\right)+\chi\left(\mathcal{L}^{-1} \otimes \mathcal{M}^{-1}\right)$.
- [x] What is the BMY inequality? ✅ 2024-05-28
	- [ ] $K^{2} \leq 9 \chi\left(\mathcal{O}_{X}\right)$
- [x] How do blowdowns affect $\rho(X)$? ✅ 2024-05-28
	- [ ] Decrease by one if the blowdown is a $(-1)$ curve.
- [x] What is a minimal surface? ✅ 2024-05-28
	- [ ] $K_{S}$ is nef, i.e., its intersection with every effective curve is non-negative.
- [x] Why can't a surface with a $(-1)$-curve be minimal? ✅ 2024-05-28
	- [ ] By adjunction, $K_S \cdot C = -1 < 0$ so $K_S$ is not nef.
- [x] What is $V(I)$ for a general $I\normal A$? ✅ 2024-05-28
	- [ ] $V(I) = \ts{\mfp \in \spec A \st \mfp \contains I}$.
	- [ ] How to remember: $V(0) = \spec A$ should be true, and every $\mfp$ contains zero. Similarly $V(A) = \emptyset$, and no prime contains all of $A$.
- [x] What is a Halphen surface? ✅ 2024-05-28
	- [ ] A smooth rational projective surface $X$ is a Halphen surface if there exists an integer $m>0$ such that the linear system $\left|-m K_{X}\right|$ is of dimension 1, has no fixed component, and has no base point. 
- [x] What is the index of a Halphen surface? ✅ 2024-05-28
	- [ ] The smallest possible value for such a positive integer $m$.
- [x] What is the classification of minimal rational surfaces? ✅ 2024-05-28
	- [ ] $\PP^2$ or $\FF_n$ for some $n\geq 0$ and $n\neq 1$, where $\FF_0 \cong \PP^1\times \PP^1$ and $\FF_1 \cong \Bl_1 \PP^2$.
- [x] What is RR for a surface? ✅ 2024-05-28
	- [ ] $h^{0}(D)+h^{0}\left(K_{X}-D\right)=h^{1}(D)+\frac{1}{2} D \cdot\left(D-K_{X}\right)+1$
- [x] What is the genus formula for a curve on a surface? ✅ 2024-05-28
	- [ ] $p_{a}(C)=1+\frac{1}{2}\left(C^{2}+C \cdot K_{X}\right)$