Download Rational Maps Between Curves and Surfaces: Properties and Relationships and more Study notes Mathematics in PDF only on Docsity!
RATIONAL MAPS BETWEEN CURVES AND SURFACES
DANIEL LOUGHRAN
- Introduction A rational map is dominant if it’s image is dense, and a rational map is birational if it is dominant and has a rational map as inverse. Finally, morphisms are rational maps which are defined on the whole variety. On general varieties, the structure of rational maps and their relationship to morphisms can be quite complicated. However, for the cases of curves and surfaces it is easier to describe. This is the aim of this lecture.
- General Properties of Rational Maps First we prove some results about rational maps in general.
Theorem 2.1 (Theorem 퐴). Let 푋 be a smooth variety, and 푓 : 푉 → ℙ푛 a rational map into projective space (e.g. into a projective variety). Then 푓 is defined everywhere except on a closed set of codimension greater than 2.
Proof. Rational maps are defined on open subsets of 푉. Let 푈 be one such open subset, where 푓 is given by
푓 : 푥 7 → (푓 0 (푥) : ⋅ ⋅ ⋅ : 푓푛(푥))
and 푓푖 ∈ 푘(푈 ) ∼= 푘(푋) are rational functions. Now by the homogeneity of projective space, we can clear denominators and highest common factors and we may assume that the 푓푖 are polynomials with gcd(푓 0 ,... , 푓푛) = 1. Now, 푓 is defined everywhere except on the set 푉 := {푥 ∈ 푈 : 푓 0 (푥) = ⋅ ⋅ ⋅ = 푓푛(푋) = 0}. This is by definition a closed set, given by more than 2 equations which have no common factors, so does not have codimension 1. □
Theorem 2.2 (Theorem 퐵). Any two varieties are birationally equivalent if and only if their function fields are isomorphic.
Theorem 2.3 (Theorem 퐶). Morphisms map projective varieties to closed sets. Note: Morphisms are continuous by definition, but they are not necessarily closed maps.
Definition 2.4. Let 휑 : 푉 1 → 푉 2 be a morphism between smooth projec- tive varieties. Then we get an induced homomorphism between the Picard Groups (here thought of as Weil Divisors modulo principal divisors)
휑∗^ : Pic(푉 2 ) → Pic(푉 1 ) 휑∗^ : 퐷 7 → 휑−^1 (퐷) 1
2 DANIEL LOUGHRAN
- Curves
The main results for curves are easy to state:
Theorem 3.1. Let 퐶 1 and 퐶 2 be non-singular projective curves over an algebraically closed field 푘 and let 휑 : 퐶 1 → 퐶 2 be a rational map. Then
(1) 휑 is in fact a morphism. (2) 휑 is either constant or surjective. (3) If 휑 is a birational map, then it is an isomorphism.
Proof. Curves contain no codimension 2 subvarieties, so (1) follows directly from Theorem 퐴. Now 퐶 1 is irreducible and closed. Since 휑 is continuous with respect to the Zariski topology and a morphism, its image is also irre- ducible and closed (Theorem 퐶). The only such subsets of 퐶 2 are 퐶 2 itself and a single point. This proves (2), and (3) is an immediate corollary. □
- Surfaces
Now on to surfaces. Throughout, let 푆, 푆 1 and 푆 2 be non-singular pro- jective surfaces over an algebraically closed field 푘.
Corollary 4.1. Any rational map between surfaces is defined everywhere except at finitely many points.
Proof. Theorem 퐴 - These are the only subvarieties of codimension 2. □
Now we shall generalize part (3) of the theorem of curves (That every birational map is an isomorphism).
4.1. Blowups and their Properties. First recall the definition of a blow-up, which is the simplest kind of birational map.
Definition 4.2. Given a surface 푆 and a point 푃 ∈ 푆, there exists a surface 푆′^ and a morphism 휋 : 푆′^ → 푆 called the blowup of 푆 at 푃 such that:
(1) The set 퐸 := 휋−^1 (푃 ) is a curve which is isomorphic to ℙ^1 , and is called the exceptional curve of the blowup. (2) The restricted map 푆′^ ∖ 퐸 → 푆 ∖ {푃 } is an isomorphism. Note: This implies that 휋 is actually a birational map.
So we are replacing a point with a copy of ℙ^1. Note that Blow ups can be explicitly constructed, but I can’t be bothered to here!
We want to determine how blowups alter the Picard Group. For this we shall appeal the following general result:
Theorem 4.3 (Theorem 퐷). Let 푉 be a smooth variety, and 푍 ⊂ 푉 a prime divisor. Then we have the following exact sequence of groups:
ℤ
푓 (^) // Pic(푉 )
푔 (^) // Pic(푉 ∖ 푍) // 0
where
푓 : 1 7 → 1 ⋅ 푍 푔 : 퐷 7 → 퐷 ∖ 푍
4 DANIEL LOUGHRAN
I also claim that we have the following commutative diagram:
Pic(푆′)
푔 (^) // Pic(푆′^ ∖ 퐸)
Pic(푆)
휋∗
O O
Pic(푆 ∖ 푃 )
To prove this, we need to show that 푔 ∘ 휋∗^ ∼= id, which we do by splitting into two cases. If 퐷 ∈ Pic(푠) is a prime divisor and 푃 ∕∈ 퐷, then
퐷
휋∗^ //휋− (^1) (퐷) 푔^ //휋− (^1) (퐷) ∖ 퐸 휋− (^1) (퐷) 퐷
since 휋−^1 (퐷) ∈ 푆′^ ∖ 퐸. Otherwise, if 푃 ∈ 퐷, then
퐷
/ /휋 휋∗−^1 (퐷 ∖ {푃 }) ∪ 퐸 푔^ //휋−^1 (퐷 ∖ {푃 }) 휋−^1 (퐷) 퐷 Hence the diagram commutes as claimed, and we have the following split short exact sequence:
0 → ℤ → Pic(푆′) → Pic(푆) → 0
thus proving the result. □
4.2. Structure of Birational maps of Surfaces. And finally the grand result on birational maps of surfaces:
Theorem 4.7. Let 휑 : 푆 1 → 푆 2 be a birational map of smooth projec- tive surfaces. Then 휑 is the composition of blowups and their (birational) inverses.
i.e. each birational map is a combination of replacing points by rational curves and replacing rational curves by points.
