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CARTIER DIVISORS
DANIEL LOUGHRAN
- Cartier Divisors The main idea behind Cartier Divisors is that any Weil divisor can be given locally by a single equation. DRAW PICTURE
Definition 1.1 (Cartier Divisor). A Cartier Divisor on a variety 푉 is a (equivalence class of) collections of pairs (푈푖, 푓푖)푖∈퐼 such that
(1) The 푈푖 are open sets which cover 푉. (2) The 푓푖 are rational functions, 푓푖 ∈ 푘(푈푖)∗^ = 푘(푉 )∗. (3) We want the functions to “glue together nicely” where they inter- sect. This is achieved as follows, for all 푖, 푗, 푓푖푓 (^) 푗− 1 is an invertible function on 푈푖 ∩ 푈푗 , i.e. 푓푖푓 (^) 푗− 1 ∈ 풪∗(푈푖 ∩ 푈푗 ). Equivalently, for all 푖, 푗, ord푈푖∩푈푗 (푓푖) = ord푈푖∩푈푗 (푓 (^) 푖− 1 ). PICTURE Two collections {(푈푖, 푓푖) : 푖 ∈ 퐼} and {(푉푗 , 푔푗 ) : 푗 ∈ 퐽} are equiva- lent (define the same Cartier Divisor) if 푓푖푔 푗− 1 ∈ 풪(푈푖 ∩ 푈푗 ), equivalently ord푈푖∩푉푗 (푓푖) = ord푈푖∩푉푗 (푔푗 ) for all 푖 ∈ 퐼, 푗 ∈ 퐽. The set of Cartier Divisor CaDiv(푉 ) forms an abelian group with the following group law:
{(푈푖, 푓푖) : 푖 ∈ 퐼} + {(푉푗 , 푔푗 ) : 푗 ∈ 퐽} := {(푈푖 ∩ 푉푗 , 푓푖푔푗 ) : (푖, 푗) ∈ 퐼 × 퐽} −{(푈푖, 푓푖) : 푖 ∈ 퐼} := {(푈푖, 푓 (^) 푖− 1 ) : 푖 ∈ 퐼} identity := {(푉, 1)}
- Examples of Cartier Divisors
Example 2.1. We will give a simple example to show how Weil Divisors correspond to Cartier Divisors. Let 푉 = ℙ^2 , with variables 푋, 푌, 푍. Consider the Cartier Divisors given by: 퐷 1 := {({푋 ∕= 0}, 1), ({푌 ∕= 0}, 푋^2 /푌 2 )} 퐷 2 := {({푋 ∕= 0}, 푌 /푋), ({푌 ∕= 0}, 1)}.
The corresponding Weil Divisors in ℙ^2 are 2 퐻 1 = 2{푋 = 0}, 퐻 2 = {푌 = 0}. Then
퐷 1 + 퐷 2 := {({푋 ∕= 0}, 푌 /푋), ({푌 ∕= 0}, 푋^2 /푌 2 ), ({푋, 푌 ∕= 0}, 1), ({푋, 푌 ∕= 0}, 푋/푌 )}
= {({푋 ∕= 0}, 푌 /푋), ({푌 ∕= 0}, 푋^2 /푌 2 )}
This corresponds to the Weil Divisor 2퐻 1 + 퐻 2. 1
2 CARTIER DIVISORS DANIEL LOUGHRAN
Definition 2.2 (Principal Cartier Divisors). Let 푓 ∈ 푘(푉 )∗. Then we define the Cartier Divisor associated to 푓 to be
Div(푓 ) = {(푉, 푓 )}. Easy definition or what?
- The Picard Group
Definition 3.1. We say two Cartier divisors are linearly equivalent if their difference is a Principal Cartier Divisor. The group of Cartier Divisors Modulo linear equivalence is called the Picard Group Pic(푉 ).
Theorem 3.2. Let 푉 be a variety whose local rings are unique factorization domains, e.g. a smooth variety. Then
CaDiv(푉 ) ∼= Div(푉 ), Pic(푉 ) ∼= Cl(푉 ).
Proof. Given a Cartier Divisor, the rational functions which occur in the ex- pression have finitely many zeros and poles which occur along subvarieties of 푉. These are well defined since the local rings are unique factorizations domains. Construct the Weil Divisor from these, with appropriate multi- plicities. This map is an isomorphism. □
Definition 3.3 (Pull-back of Divisors). Let 푔 : 푋 → 푌 be a morphism of varieties. Let 퐷 = {(푈푖, 푓푖) : 푖 ∈ 퐼 ∈ CaDiv(푌 )}. Then we can define the Pull-back of 퐷 by 푔 on 푋 to be
푔∗(퐷) := {(푔−^1 (푈푖), 푓푖 ∘ 푔) : 푖 ∈ 퐼}.
It can be checked that this gives us a homomorphism: 푔∗^ : Pic(푌 ) → Pic(푋). Note for Joe: Pic(⋅) is a contravariant functor!
Example 3.4 (Picard Group of Projective Varieties). If your variety is pro- jective, you get lots of Divisors for “free”, which come from hypersurfaces. Let 푉, 푋 ⊂ ℙ푛^ be hypersurfaces, such that 푋 is not contained in 푉 , and with 푋 = 푉 (퐹 ) and deg(퐹 ) = 푑. Cover 푉 be the open affine subsets 푈푖 := 푉 ∖ {푥푖 = 0}. (Note: any choice of open cover will suffice, but we choose the easiest one). Then define
Note that the 1/푥푑푖 are just fudge factors, since the 푥푖 don’t have any zeros or poles on the open sets we have defined! If 퐺 is another homogeneous polynomial of degree 푑, then
(퐹 )푉 − (퐺)푉 = {(푈푖 ∩ 푈푗 , (퐹/푥푑푖 )(퐺/푥푑푗 )−^1 ) : 0 ≤ 푖, 푗 ≤ 푛} = {(푉, 퐹/퐺)} = Div(퐹/퐺).
Hence (퐹 )푉 ∼ (퐺)푉. So any two hypersurfaces in ℙ푛^ of the same degree induce the same element of Pic(푉 ).
