---
sort: 020
title: "Varieties: Definitions"
flashcard: "Orals::Varieties, Definitions"
---

# Varieties

- [x] For $V$ a affine variety, what is $k[V]$? What is $A(V)$? What is $k(V)$? ✅ 2022-11-23
	- [ ] For $V\subseteq \AA^n$, these are polynomial functions $f\in k[x_1,\cdots, x_n]$ restricted to $V$, i.e. $k[V] = k[x_1,\cdots, x_n]\mid_V$. This can be written as $k[V] = k[x_0,\cdots, x_n]/I(V)$, and is sometimes written $A(V) \da k[V]$. 
	      Then $k(V)$ are rational such functions.
- [x] What is a **Noetherian** space? ✅ 2022-11-23
	- [ ] DCC on closed subsets (not necessarily irreducible)
- [x] What is ${\mathcal{O}}_X$ for an affine variety? ✅ 2022-12-04
	- [ ] The sheaf of regular functions: sections over $U$ are functions $\phi: U\to k$ where for every $p\in U$ there exists an open neighborhood $U_p\ni p$ and polynomials $f, g\in k[X]$ with $\ro{\phi(x)}{U_p} = {f(x)\over g(x)}$.
- [x] How is ${\mathcal{O}}_X$ defined for an *affine scheme* $\spec A$? ✅ 2022-12-04
	- [ ] For $X = \spec A$, define $\OO(U)$ to be functions $\phi: U\to \coprod_{p\in U} A_p$ such that $\phi(p) \in A_p$ for every $p$ and $s$ is locally a quotient, i.e. for every $p$ there is a $V_p\ni p$ and elements $f, g\in A$ such that for each $q\in V_p$, with $f\not\in q$ and $\phi(q) =  {f\over g}$.
- [x] What are the **Zariski closed** subsets of ${\mathbb{A}}^n$? ✅ 2022-11-27
	- [ ] For $\AA^1$ (and any irreducible curve), the Zariski topology on the closed points is the cofinite topology. For $\AA^n$, not cofinite, just take the usual basis $V(I)$ for $I\normal \kxn$.
- [x] Is the Zariski topology on $\AA^2$ the product topology? ✅ 2022-12-04
	- [ ] No: $V(x-y) \subseteq \AA^2$ is clearly closed in the Zariski topology but not closed in the product topology. If it were, $\AA^2\sm\Delta$ would be open, so find $U\times V\subset \AA^2\sm\Delta$ an open box. Then $U,V$ have finite complements since $\AA^1$ has the cofinite topology, $U = \AA^1\sm\ts{p_i}_{i\leq n_1}$ and $V = \AA^1\smts{q_i}_{i\leq n_2}$. Pick any $z\neq p_i, q_i$, then $(z,z)\in U\times V \intersect \Delta$.
- [x] What is a [distinguished open set](distinguished%20open%20set)? ✅ 2022-11-27
	- [ ] $D_f \da X\sm V(f)$.
- [x] What is an **affine variety**? A **projective variety**? ✅ 2022-11-27
	- [ ] Affine: an irreducible closed subset of $\AA^n$. Projective: an irreducible algebraic subset of $\PP^n$, where $Y$ is an algebraic subset if $Y = V(T)$ for $T$ a collection of homogeneous elements in $k[x_0,\cdots, x_n]$.
- [x] What is an **irreducible** subset of an affine variety? ✅ 2022-11-27
	- [ ] Can't be expressed as the union of proper closed subsets.
- [x] What is a **proper** variety? ✅ 2022-11-27
	- [ ] Separated, finite type, universally closed. Can leave out finite type, since all varieties are.
- [x] Give an example of a non-proper variety. ✅ 2022-12-04
	- [ ] $\AA^n$: separated and finite type over $k$, but not universally closed: take $X = \AA^1, Y= \AA^1$, then $X\fiberprod{k} Y$ is the projection $\AA^2\to \AA^1$ onto the coordinate axis, but e.g. $V(xy-c)\mapsto \GG_m$ which is not closed.
- [x] What is a [complete](complete.md) variety? ✅ 2022-11-27
	- [ ] Integral separated scheme of finite type over a field $k$, where $X\to \spec k$ is proper.
- [x] What is a [[normal]] affine variety? ✅ 2022-11-27
	- [ ] Irreducible and local rings are integrally closed.
- [x] What is a [Fano](Unsorted/Fano%20variety.md) variety? ✅ 2022-11-27
	- [ ] A complete variety with ample anticanonical $-K_X$. Note: always projective.
- [x] What is a del Pezzo surface? ✅ 2022-11-27
	- [ ] Fano of dimension 2.
- [x] Give several examples of Fano varieties. ✅ 2022-12-04
	- [ ] $-K_{\PP^n} = \OO(n+1)$ which is very ample.
	- [ ] Any $D\in \Div(\PP^n)$ when $\deg D < n+1$, since adjunction yields
	      $$K_D = (K_{\PP^n} + D)\mid_D = \qty{-(n+1)H + \deg(D)\cdot H}\mid_D > 0$$
- [x] What is a [Calabi-Yau](Unsorted/Calabi-Yau.md) variety? ✅ 2022-12-04
	- [ ] $K_X \sim_\QQ 0$. Sometimes require $\omega_X\cong \OO_X$ instead, with $h^j(\OO_X) = 0$ for $1\leq j\leq n-1$.
- [x] Give examples of Calabi-Yau varieties. ✅ 2022-12-04
	- [ ] Hypersurfaces $D$ of $\deg D = n+1$, since adjunction yields
	      $$K_D = (K_{\PP^n} + D)\mid_D = \qty{-(n+1)H + \deg(D)\cdot H}\mid_D = 0H\mid_D = 0,$$
	      using the Lefschetz hyperplane theorem to show that the $h^i$ vanish.
- [x] What is the **dimension** of an affine variety? ✅ 2022-11-27
	- [ ] Supremum over all $n$ of lengths of chains $C_0 \subseteq \cdots C_n$ of irreducible closed subsets.
- [x] What is the **degree** of a variety? ✅ 2022-11-27
	- [ ] For $\dim X = n$, the number of points in $X$ intersected with $n$ general hyperplanes.
- [x] What is a **smooth point** of a variety? A **singular point**? ✅ 2022-11-27
	- [ ] Jacobian criterion: let $I(Y) = \gens{f_1,\cdots, f_r} \normal \kxn$ and $J_Y \da \left[ \dd{f_i}{x_j} \right]$ the corresponding Jacobian matrix, then $Y$ is smooth at $p$ iff $\rank J_Y(p) = n-r$ (maximal rank), otherwise if $\rank J_Y(p) < n-r$ it is a singular point.
	- [ ] Alternatively, $\dim \T_p X = \dim X$ if smooth and $\dim \T_p X > \dim X$ if singular, where $\T_p X \da (\mfm_p /\mfm_p^2)\dual  \da \Hom_{\kmod}(\mfm_p/\mfm_p^2, k)$.
- [x] What is a [separated](separated.md) variety? ✅ 2022-11-27
	- [ ] Separated iff diagonal is closed iff at $\leq 1$ lift exists for the valuative criterion (to exclude double points).
- [x] What is the tautological bundle on $\PP^n$? ✅ 2022-11-27
	- [ ] $\OO_{\PP^n}(-1)$, the bundle whose fiber above $p\in \PP^n$ is the line in $\AA^{n+1}$ defined by $p$.
- [x] What is the **Serre twisting line bundle**? ✅ 2022-12-04
	- [ ] The dual of the tautological, $\OO(-1)\dual = \OO(1)$.
- [x] How can you check if $X = V(I)$ is irreducible? ✅ 2022-12-04
	- [ ] Iff $I$ is prime in $\kxn$.
- [x] Why does the exceptional curve $E$ in $\Bl_p(X)$ satisfy $E^2=-1$? ✅ 2022-11-30
	- [ ] Take $\pi:\Bl_0 \PP^2\to \PP^2$. Take two lines $L_1, L_2$ intersecting transversally away from 0 in $\PP^2$, then $\pi^* L_1.\pi^* L_2 = L_1.L_2 = 1$ since $\pi$ is an isomorphism away from zero.
	      Now move them to $L_1', L_2'$ intersecting at 0; then $\pi^* L_1' = \tilde L_1' + E$ and $\pi^* L_2' = \tilde L_2' + E$, and 
	      $$1 = (\pi^*L_1' . \pi^* L_2') = (\tilde L_1' + E).(\tilde L_2' + E) = \tilde L_1'.\tilde L_2' + \tilde L_1'.E + E.\tilde L_2' + E^2 = 0 + 1 + 1 + E^2 \implies E^2=-1$$.
	- [ ] Use the following theorem:
	      ![](attachments/Pasted%20image%2020221204223854.png)
	      For $\pi: \Bl_p X\to X$, let $C$ be a curve in $X$, smooth at $p$.
	      Then the proper transform $\tilde C$ intersects $E$ transversely at one point $q$, and $\pi^* C = \tilde C + E$.
	      Intersect both sides with $E$ to get $\pi^* C . E = (\tilde C + E).E$. 
	      The LHS is zero since $\pi^* A.E = 0$ for any divisor $A\in X$. 
	      The RHS is $\tilde C.E + E^2 = q + E^2$; taking degrees yields $0 = 1 + E^2 \implies E^2 = -1$.
- [x] Why is the normalization a resolution of singularities in dimension 1? ✅ 2022-11-30
	- [ ] The normalization is normal, hence regular in codimension 1, hence regular hence smooth.
- [x] What is the canonical class of a blowup? ✅ 2022-12-04
	- [ ] For $\pi: \tilde X\to X$, $K_{\tilde X} = \pi^* K_X + E$.