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ALGEBRAIC GEOMETRY I, FALL 2016. 1. Weil and Cartier Divisors

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ALGEBRAIC GEOMETRY I, FALL 2016. 1. Weil and Cartier Divisors ALGEBRAIC GEOMETRY I, FALL 2016. DIVISORS. 1. Weil and Cartier divisors Let X be an algebraic variety. Define a Weil divisor on X as a formal (finite) linear combination of irreducible subvarieties of codimension 1 with integral coefficients: m P D = aiDi, where ai 2 Z. Define a Cartier divisor on X as a collection of rational i=1 S functions fi on Ui, where X = Ui is an open cover, such that fi=fj has no zeroes or poles on Ui \ Uj, up to the following equivalence relation: we say that f(Ui;Fi)g is equivalent to f(Wj; gj)g if the covers fUig and fWjg have a common refinemenent fZkg, and if Zk ⊂ Ui, Zk ⊂ Wj, then (fijZk )=(gjjZk ) has neither zeroes nor poles on Zk. Definition 1. A principal divisor is a Cartier divisor that is defined by a single rational function f on X. On a reasonably good variety (such as a smooth variety) we can define the order of a rational function f along a subvariety Y ⊂ X of codimension 1. If X is the affine line k1, Y is a point, and f = P=Q is a rational function, the order of f at (along) Y is k if P has a zero at Y with multiplicity k, and −k if Q has a zero at Y with multiplicity k. Once we have this notion, we can define a Weil divisor for any principal Cartier P divisor by taking the sum aiXi over all subvarieties Xi ⊂ X of codimension 1 where ai is the order of the function f along Xi. This sum will be finite. Then we can define a Weil divisor for any Cartier divisor f(Ui; fi)g by taking the above sum on every Ui and noticing that these Weil divisors agree on the intersections Ui \ Uj. On smooth varieties, Weil divisors are in bijection with Cartier divisors. On singular varieties, there may be Weil divisors that cannot be given as Cartier divisors, or non-trivial Cartier divisors for which the operation above produces a zero Weil divisor. Weil divisors naturally form an abelian group (we just add the linear combinations formally). Cartier divisors also form an abelian group: for two Cartier divisors we can take a common refinement of the open covers and take the product of the functions defining the divisors. It is easy to see that the map from Cartier to Weil divisors described above is a group homomorphism. Definition 2. Two divisors (Weil or Cartier) are called linearly equivalent if their difference is a principal divisor. The group of all Cartier divisors on X modulo linear equivalence is called the Picard group of X and denoted by Pic(X). 1 2 DIVISORS. 2. Cartier divisors and line bundles The data of a Cartier divisor in fact consists of the data of a line bundle with a rational section: having f(Ui; fi)g take the bundle that is trivial over each Ui, and define the transition functions as gij = fj=fi (regular function on Ui \Uj). Then ffig define a rational section. A sum of two divisors corresponds to the tensor product of their respective ine bundles. Lemma 1. Two Cartier divisors are linearly equivalent if and only if the line bundles that correspond to them are isomorphic. Proof. Suppose we have two sections f(Ui; si)g and f(Wj; tj)g of the same line bundle. By choosing a common refinement, we can assume that the coverings fUig and fWig are the same. Then on each open set si=ti defines a rational function, and these functions coincide on the intersections Ui\Uj, so they define a global rational function that gives the linear equivalence between the corresponding Cartier divisors. On the other hand, if two Cartier divisors are linearly equivalent, we can again assume that they are described by functions on the same covering: f(Ui; fi)g and f(Ui; gi)g with fi = fgi for some rational function f on X. Then the corresponding line bundles have transition functions gi=gj and fi=fj = fgi=fgj = gi=gj, which gives the same line bundle. As a corollary, we get an alternative definition of Pic(X) as the group of line bundles on X with tensor product as the operation. n Example 1. We have established that every line bundle on P is of the form O(k) for some k 2 Z. Since O(k) ⊗ O(m) 'O(k + m), we have an isomorphism of abelian n groups Pic(P ) ' Z. 3. Linear systems P Definition 3. A divisor D = aiDi on an algebraic variety X is effective if ai > 0 for all i. Let X be a smooth variety, and let D be an effective divisor on it. Then D is the divisor of a regular section s of the line bundle O(D). Any other effective divisor D0 that is linearly equivalent to D is the divisor of another section s0 of O(D). These sections are defined uniquely up to multiplication by a non-zero scalar, and each section of O(D) defines an effective divisor. The space Γ(O(D)) of all global regular sections of O(D) is a vector space, so effective divisors that are linearly equivalent to D are in bijection with its projectivization P(Γ(O(D))). Definition 4. A complete linear system jDj of divisors on X is the set of all effective divisors that are linearly equivalent to a given divisor D, viewed as a projective space P(Γ(O(D))) (the divisor D itself does not have to be effective). A linear system of divisors is a projective subspace of a complete linear system, i.e. a projectivization of a linear subspace in Γ(O(D)) for some D. n Examples of complete linear systems are the space of all hyperplanes in P , the 2 space of all quadric curves on P , etc. An example of a linear system that is not a 2 complete linear system would be all quadric curves on P passing through a given point. ALGEBRAIC GEOMETRY I, FALL 2016. 3 Definition 5. The base locus of a linear system of divisors is the set of points such that all of the divisors in the linear system pass through them. We say that the line bundle O(D) is generated by global sections if for any point x 2 X there is a section s 2 Γ(O(D) such that s(x) 6= 0; equivalently, O(D) is generated by global sections if the base locus of the linear system jDj is empty. A finite collection of sections s0; : : : ; sN of a line bundle L on X such that for N every x 2 X there is i such that si(x) 6= 0 determines a map X ! P . Indeed, let fUig be a covering of X such that L is trivial on every Ui. Each sk is given by a collection of functions fik on Ui. Then to each x 2 Ui we can associate the N point (fi0(x): ::: : fiN (x)) in P . This point is well-defined because all these values cannot be zero simutaneously, and it does not depend on i because for every k the values fik(x) and fjk(x) differ by the same scalar factor gij(x), where gij are N transition functions for L. The space P here is naturally the dual space to the projectivization of the space spanned by si themselves. In the case where D is a divisor such that O(D) is generated by global sections _ and dim Γ(O(D)) < 1, we get a map X ! P(Γ(O(D))) . 1 Example 2. If D is a divisor of degree d on P , then the map defined by jDj is the 1 vd d Veronese embedding P −! P . n Example 3. If X ⊂ P and D is the intersection of X and a hyperplane (equiv- alently, O(D) is the restriction of O(1) to X), then the map defined by jDj is the n embedding X ! P that we started with. π n n Example 4. If X −! P is the blow-up of a point p in P and D is the preimage of n a hyperplane in P that does not contain p, the map defined by jDj is π. 4. Ample and very ample divisors So far we have established that any linear system with empty base locus defines a map from X to a projective space. This map is not always injective (see example of the blow-up above), but we are especially interested in the divisors whose complete linear system does define an injective map. _ Definition 6. A divisor D on X is very ample if the map X ! P(Γ(O(D))) is an embedding. Equivalently, D is very ample if the line bundle O(D) is isomorphic to N N the restriction of the line bundle O(1) from P to X for some embedding X ⊂ P . A divisor D is ample, if mD is very ample for some m > 0. n Example 5. For X = P and a line bundle L'O(k) the following are equivalent: L is ample, L is very ample, L is generated by global sections, k > 0. Example 6. On a smooth projective curve X, a divisor D is ample if and only if its degree is positive. This follows from the Riemann-Roch Theorem. We will not give the proof for the following theorem and proposition.
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