# Varieties

# Quasi-Projective Varieties

# Quasi-Projective Varieties

A quasi-projective variety is a locally closed subset of Pn\mathbb{P}^{n}, considered witht the Zariski topology induced from Pn\mathbb{P}^{n}. A locally closed set of any topological space is a closed subset of an open subspace, that is, an intersection of an open set and a closed set.

Examples

The class of quasi-projective varieties includes all projective varieties, all affine varieties, and all Zariski open subsets of these.

# Morphisms

Let VPnV \subseteq \mathbb{P}^{n} and WPmW \subseteq \mathbb{P}^{m} be quasi-projective varieties. A morphism F:VWF: V \rightarrow W is a map such that for each pVp \in V, there exist m+1m+1 homogeneous polynomials F0,,FmF_{0},\ldots, F_{m} in n+1n+1 variables such that the map

VPm,q[F0(q)::Fm(q)],V ⟶ \mathbb{P}^{m},\qquad q ⟼ [F_{0}(q): \ldots : F_{m}(q)],

is well-defined at pp and agress with FF on some non-empty open set containing pp.

# Birational Geometry

# Projective Morphisms

A morphism of varieties π:XV\pi : X \rightarrow V is called a projective morphism if XX is a closed subvariety of a product variety XV×PnX \subset V \times \mathbb{P}^{n} and π:XV\pi : X \rightarrow V is the restricion of the projection onto the first coordinate.

Remark

Projective morphisms have the property that the preimage of any point is a projective variety. A projective morphism is a proper morphism in the Euclidean topology.

# Birational Morphisms

A morphism π:XV\pi : X \rightarrow V of quasi-projective varieties is called a birational morphism if its restriction to some dense open set UXU \subset X is an isomorphism onto some dense open subset UVU^{\prime} \subset V.

Example

The blow-up of a point in nn-dimensional affine space An\mathbb{A}^{n} is an example of a birational morphism.

# Rational Maps

Let XX be a quasi-projective variety, and let UU and UU^{\prime} be dense open subsets of XX. Suppose we are given two morphisms φ:UY\varphi : U \rightarrow Y and φ:UY\varphi^{\prime} : U^{\prime} \rightarrow Y of quasi-projective varieties. We say that (U,φ)(U,\varphi) and (U,φ)(U^{\prime}, \varphi^{\prime}) are equivalent if the mappings φ\varphi and φ\varphi^{\prime} coincide on the intersection UUU \cap U^{\prime}.

A rational map XYX ⇢ Y is an equivalence class of morphisms defined on dense open sets of XX as above.

Example

Projecting from a point in projective space is an example of a rational map. Let HPnH \subset \mathbb{P}^{n} be a fixed hyperplane in Pn\mathbb{P}^{n} and let pPnp \in \mathbb{P}^{n} be any point not on HH. The projection from pp onto the hyperplane HH is the rational map

φ:PnH=Pn1,xφ(x):={the unique intersection point of H and the line xp}.\varphi : \mathbb{P}^{n} ⇢ H = \mathbb{P}^{n-1},\qquad x ⟼ φ(x) := \{ \text{the unique intersection point of $H$ and the line $\overline{xp}$} \}.

The rational map is a well-defined morphism everywhere outside pp.

# Birational Equivalence

Let XX and YY be irreducible algebraic varieties. They are called birationally equivalent if there a re mutually inverse rational maps

F:XYandG:YX.F: X ⇢ Y\qquad \text{and} \qquad G: Y ⇢ X.

Remark

This