ARITHMETIC PROPERTIES OF DEL PEZZO SURFACES DAMIANO TESTA Written mental notes. These are just some rough notes meant as written mental notes of what the course will cover. Introduction In this series of lectures, I will present some results on arithmetic properties of del Pezzo surfaces over a field k. I will try to impose as few restrictions as possible on k. Important fields to keep in mind are • the complex numbers C; • algebraically closed fields; • finite fields Fq; • p-adic fields Qp and their finite extensions; • number fields. I will recall and sometimes give proofs of facts on del Pezzo surfaces over alge- braically closed fields. I will then take these facts as our guiding principles to figure out what can be said over a field that is not necessarily algebraically closed. Here is an outline of the course. We begin by studying the exceptional curves on del Pezzo surfaces. Next, we focus on del Pezzo surfaces of degree at least 7. The analysis shows that given a k-rational point on a del Pezzo surface, we can often produce more k-rational points. This naturally leads to the question of k- (uni)rationality for del Pezzo surfaces. We will see that del Pezzo surfaces of certain degrees automatically have a k-rational point. Finally, we may touch upon Cox rings of del Pezzo surfaces, universal torsors and Manin's conjecture. 1. Background and the case of curves Definition 1.1. Let k be a field and let X be a scheme over k.A k-rational point on X is a morphism Spec k ! X. We denote the set of all k-rational points of X by X(k). n If X ⊂ Ak is affine and it is the vanishing set of polynomials f1; : : : ; fr 2 n k[x1; : : : ; xn], then a k-rational point on X is an n-tuple x = (x1;:::; xn) 2 k such that f1(x) = ··· = fr(x) = 0. Lemma 1.2 (Lang-Nishimura). Let k be a field, let X; Y be varieties over k and let ': X 99K Y be a rational map. If X has a smooth k-rational point and Y is proper, then Y has a k-rational point. Proof. Proceed by induction on the dimension of X. If the dimension of X is 0, then do the obvious. For the induction step, use the valuative criterion of properness to show that a rational map from X to Y determines a rational map from any divisor Date: July 2019. 1 2 DAMIANO TESTA on X to Y . Choose a divisor containing a smooth k-rational point of X as a smooth point and conclude by induction. An immediate consequence of the Lang-Nishimura Lemma is that having a k- rational point of a birational invariant among smooth proper varieties. Corollary 1.3. If X and Y are smooth proper varieties defined over a field k, then X has a k-rational point if and only if Y has a k-rational point. Proof. Use the Lang-Nishimura Lemma and the symmetry of the assumptions. Let k be a field and let X be a smooth variety of dimension n defined over k. We denote by !X a dualizing sheaf on X and we call it the canonical line bundle. Vn Thus, the sheaf !X is a line bundle isomorphic to ΩX , the determinant of the sheaf of 1-forms on X. We shall mostly use this for curves and surfaces, i.e. for n ≤ 2. Definition 1.4. A Fano variety over k is a smooth projective variety X defined _ over k with ample anti-canonical line bundle (!X ) . The degree dX of X is the d positive integer dX = (−KX ) . In this course, we will mostly be interested in the case of Fano varieties of dimension 2, that is, del Pezzo surfaces, over general fields. In our arguments, we will often switch between a field k and an extension of k, often a fixed algebraic closure k. In doing so, we will want to keep track of what changes and what stays constant. One of the most basic facts that we will use is the following result. Fact. Let k0 ⊃ k be an extension of fields. Let X be a projective variety over k and let L be a line bundle on X. Denote by Lk0 the pull-back of L to the base change 0 0 0 0 Xk0 of X to k . The k -vector space H (Xk0 ; Lk0 ) is isomorphic to the k -vector 0 0 space H (X; L ) ⊗k k . The more general result with global sections replaced by an arbitrary cohomology group of a quasi-coherent sheaf on X is also true (see Cohomology and base-change). Nevertheless, the mentioned fact is enough for most of our applications. Indeed, we can almost get away simply with knowing that the dimensions of the spaces of global sections of line bundles are constant when extending the base field. ARITHMETIC PROPERTIES OF DEL PEZZO SURFACES 3 Aside. Let us pause to think about \base rings". Suppose that X is the vanishing set of an ideal I in k[x1; : : : ; xn]. There are at least three \base rings" that can, and will, be relevant for this course. (Tautological) Of course, we defined I to be an ideal in k[x1; : : : ; xn], so certainly k is a possible \base ring". (Geometric) We may want to allow ourselves to pick coordinates of points on the inter- section of X with curves or surfaces, to extract coefficients of equations of special curves on X, or more generally to be able to make finite extensions of k. Thus, certainly also the collection of all finite extensions of k, i.e. an algebraic closure k of k, is a possible \base ring". (Minimal) We may only want to look at the smallest subfield (or even the smallest subring) k0 of k containing the coefficients of a set of generators of I. Again, k0 is another good candidate for a \base ring". Really, we should make up our mind on what the base ring is and stick to the tau- tological option. For instance, we may be in a situation in which X is defined over the complex numbers by an equation with integer coefficients, and we want to reduce modulo 2 to deduce properties over C. In such a situation, we may (re-)define a sur- face over Spec Z with the same equation as X and we can now base change using Spec F2 ! Spec Z or Spec C ! Spec Z. Before looking at the surface case, we give a quick overview of what happens in the case of Fano curves. If the field k is algebraically closed, then a smooth projective curve over k with ample anti-canonical line bundle (that is, ample tangent bundle) is isomorphic 1 to Pk. Thus, we have a full classification, with only one geometric example. Now, let C be a smooth projective Fano curve over an arbitrary ground field k. _ _ _ The anti-canonical linear system j!C j on C induces a rational map C 99K j!C j . By the previously mentioned Cohomology and base-change, we know that the linear _ 2 system j!C j has dimension 2 and hence we obtain a rational map ι: C 99K Pk. We fix an algebraic closure k of k and base-change to k. The curve C becomes isomorphic 1 _ to P , the line bundle j!C j becomes isomorphic to O 1 (2) and the rational map ι k Pk becomes an isomorphism betwen 1 and a smooth plane conic. We deduce that ι Pk induces an isomorphism between the curve C and a smooth plane conic. 1 Observe that Pk is certainly a Fano curve, regardless of whether or not the field k 1 is algebraically closed. The following exercise gives a characterisation of Pk among Fano curves. Exercise 1.5. Let C be a Fano curve over a field k. The following are equivalent. 1 (1) The curves C and Pk are isomorphic over k. (2) There is a k-rational point on C. (3) There is a line bundle on C of degree 1. (4) There is a line bundle on C of odd degree. [Hints. (1) =) (2), use the Lang-Nishimura Lemma. (2) =) (1), realize C as a conic and use stereographic projection. (2) =) (3) =) (4), easy. (3) =) (2), the vanishing set of any global section gives you a point. (4) =) (3), using a line bundle of odd degree and the canonical line bundle, produce a line bundle of degree 1. ] 1 From the previous exercise, if C is isomorphic to Pk over the field k, then C has at least one k-rational point. We can therefore easily find examples of Fano curves 1 that are not isomorphic to Pk. 4 DAMIANO TESTA Exercise 1.6. The conic C ⊂ 2 defined by the equation x2 + y2 + z2 = 0 is a PQ Fano curve that is not isomorphic to 1 . PQ We are therefore naturally led to examine the question of whether or not a variety has a rational point. Clearly, if a variety defined over a field k has a k-rational point, then it also has a point over every field extension k0 of k. In the case in which the ground field is Q, there are some natural extensions of Q that play an important role. For instance, the real numbers R are the completion of Q with respect to the familiar Euclidean absolute value on Q.