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Introduction to Projective Varieties by Enrique Arrondo(*)

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Introduction to Projective Varieties by Enrique Arrondo(*) Introduction to projective varieties by Enrique Arrondo(*) Version of November 26, 2017 This is still probably far from being a final version, especially since I had no time yet to complete the second part (which is so far not well connected with the first one). Anyway, any kind of comments are very welcome, in particular those concerning the general structure, or suggestions for shorter and/or correct proofs. 0. Algebraic background 1. Projective sets and their ideals; Weak Nullstellensatz 2. Irreducible components 3. Hilbert polynomial. Nullstellensatz 4. Graded modules; resolutions and primary decomposition 5. Dimension, degree and arithmetic genus 6. Product of varieties 7. Regular maps 8. Properties of morphisms 9. Resolutions and dimension 10. Ruled varieties 11. Tangent spaces and cones; smoothness 12. Transversality 13. Parameter spaces 14. Affine varieties vs projective varieties; sheaves 15. The local ring at a point 16. Introduction to affine and projective schemes 17. Vector bundles 18. Coherent sheaves 19. Schemes (*) Departamento de Algebra,´ Facultad de Ciencias Matem´aticas, Universidad Com- plutense de Madrid, 28040 Madrid, Spain, Enrique [email protected] 1 The scope of these notes is to present a soft and practical introduction to algebraic geometry, i.e. with very few algebraic requirements but arriving soon to deep results and concrete examples that can be obtained \by hand". The notes are based on some basic PhD courses (Milan 1998 and Florence 2000) and a summer course (Perugia 1998) that I taught. I decided to produce these notes while preparing new similar courses (Milan and Perugia 2001). My approach consists of avoiding all the algebraic preliminaries that a standard al- gebraic geometry course uses for affine varieties and thus start directly with projective varieties (which are the varieties that have good properties). The main technique I use is the Hilbert polynomial, from which it is possible to rigorously and intuitively introduce all the invariants of a projective variety (dimension, degree and arithmetic genus). It is also possible to easily prove the projective Nullstellensatz (from which the standard affine Nullstellensatz can in fact be obtained). The price to pay for this shortcut is that the way to produce the important results (the most important one for practical purposes is the theorem about the dimension of the fibers) is not always clear, since many results or even definitions have local nature. This was in fact the eventual motivation to write these notes, to show that it is possible to follow such a risky path in a coherent way. Moreover, if the students of a course have all the delicate steps written down, it is possible for the teacher to avoid the too technical results and concentrate on examples and intuitive results. If the goal of theses notes is achieved, an interested student with very small knowledge of commutative algebra (a sight to Chapter 0, devoted to preliminaries should be enough to figure out the required background) should be able to acquire enough techniques to ma- nipulate varieties and families and compute their dimensions. And if the student becomes interested, he/she could then follow a more advanced text or course. The present version contains the beginning of a second part which, if ever finished, will eventually contain a first introduction to the theory of schemes. I deeply thank many people, and very especially Sof´ıaCobo, for pointing out many misprints of previous versions. 2 0. Algebraic background We will recall in this chapter the main algebraic ingredients that the reader is as- sumed to know as a minimum background for the rest of the notes. We will just give the appropriate definitions, state the results and leave the proofs (for those for whom they are new concepts) as exercises, as long as it looks reasonable to do so. The key results the reader should know about ideals are collected in the following exercise. Exercise 0.1. Let R be a (commutative and unitary) ring and I an ideal. (i) If I is maximal then I is prime. p (ii) The set I := fF 2 R j F d 2 I for some d 2 Ng is an ideal of R. p (iii) If I is prime, then it is radical (i.e. I = I). p p p (iv) If I = I1 \ ::: \ In then I = I1 \ ::: \ In. (v) If I is prime and I1 \ ::: \ In ⊂ I then Ii ⊂ I for some i = 1; : : : ; n. (vi) If I is an ideal contained in a finite union of prime ideals, then I is contained in one of those prime ideals. p (vii) If I is any ideal of R, then I is the intersection of the prime ideals containing I. p [Hint: If f 62 I, use Zorn's Lemma to find a maximal element of the set of ideals p J ⊃ I such that f 62 J, and prove that such a maximal element is a prime ideal] (viii) If R0 is another ring, f : R0 ! R is a ring homomorphism (we will always assume that a ring homomorphism sends the unit element of R0 to the unit element of R) and I is a prime ideal of R, then f −1(I) is a prime ideal of R0. Definition. A primary ideal of a ring R is an ideal I with the property that if FG 2 I but G 62 I then there exists some d 2 such that F d 2 I. It is immediate to see that if I p N is primary, then P := I is a prime ideal. The ideal I is then said to be P -primary. Exercise 0.2. Let I be an ideal of a ring R. p (i) Prove that if I is primary then I is prime. (ii) Find a counterexample showing that it is not true that an ideal whose radical is prime is necessarily primary (the reader should be able to produce many examples after section 2). p (iii) If I is a maximal ideal, prove that I is a primary ideal. (iv) If I = \iIi where each Ii is a P -primary ideal, then I is also P -primary. (v) If R is a polynomial ring and I is generated by f m, f being an irreducible polynomial, then I is (f) − primary. 3 (vi) If R0 is another ring, f : R0 ! R is a ring homomorphism and I is P -primary, then f −1(I) is f −1(P )-primary (observe that from Exercise 0.1(viii) f −1(P ) is a prime ideal). We are going to work with polynomial rings over a field. The main result about these rings is that all their ideals can be generated by a finite number of elements. This will be a consequence of the so-called Hilbert's bases theorem, which we will prove below. Definition. A ring R is called a noetherian ring if any ideal of I admits a finite number of generators, or equivalently if R does not contain an infinite strictly ascending chain of ideals I1 ⊆= I2 ⊆= :::. Exercise 0.3. Prove that indeed the above two definitions are equivalent. [Hint: Observe that the union of all the ideals in an ascending chain is an ideal]. Theorem 0.4 (Hilbert's basis theorem). Let R be a noetherian ring. Then the polynomial ring R[X] is noetherian. Proof: Let I be an ideal of R[X]. We can assume I 6= R[X], since otherwise 1 would be a generator of I. For each d 2 N, the set Jd := fr 2 R j r is the leading coefficient of some polynomial of degree d in Ig is easily seen to be an ideal of R (if we take the convention that 0 2 Jd) and J1 ⊂ J2 ⊂ :::. Since R is noetherian, there exists d0 2 N such that Jd = Jd0 if d ≥ d0. On the other hand, we can find polynomials f1; : : : ; fm 2 I such that each J0;:::;Jd0 is generated by the leading coefficients of some (not necessarily all) of these polynomials. Let us see that these polynomials generate I. Take f 2 I and let d be its degree. Assume first that d ≥ d0. Then the leading coefficient of f is a linear combination (with coefficients in R) of the leading coefficients of d−deg fi f1; : : : ; fm. Multiplying each fi by X we see that there exist monomials h1; : : : ; hn 2 R[X] such that f − h1f1 − ::: − hnfn (which is still in I) has degree strictly less than d. Iterating the process we arrive to g1; : : : ; gn 2 R[X] such that f−g1f1−:::−gnfn has degree strictly less that d0. Hence we can assume d < d0. But since now Jd is generated by some leading coefficients of f1; : : : ; fn, we can find r1; : : : ; rn 2 R such that f − r1f1 − ::: − rnfn has degree strictly smaller than d. Iterating the process till degree zero we then find that it is possible to write f as a linear combination of f1; : : : ; fn , which concludes the proof. Exercise 0.5. Prove a stronger result in case R is a field, namely that any ideal in K[X] is principal, i.e. generated by one polynomial (a ring with this property is called a principal ideal domain or PID for short). [Hint: Consider a nonzero polynomial of minimum degree 4 of an ideal and prove, dividing by it, that any other polynomial of the ideal is a multiple of it]. Exercise 0.6. Prove, using induction on n and the Hilbert's basis theorem, that the polynomial ring K[X1;:::;Xn] is noetherian. Exercise 0.7. Prove that, for any ideal I ⊂ K[X1;:::;Xn], there exists some m 2 N such p m that I ⊂ I (in fact this is true for any ideal of a noetherian ring).
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