Lecture 21: Riemann-Hurwitz Formula, Chevalley’s Theorem

Lecture  18.725 Algebraic Geometry I Lecture 21: Riemann-Hurwitz Formula, Chevalley’s Theorem We begin with a remark on the tangent cone. Let X be a variety and x ∈ X. ˆ → X is • We checked that Spec((⊕mnx /mxn+1 )red ) is the tangent cone over π −1 (x) ⊂ Pn , where π : X the blow-up of X at x. If X = Spec A we can do this for any ideal in A; indeed, applying it to IZ , where Z ⊂ X is a closed subvariety, we get that the “normal cone” to Z is Spec ((⊕IZn /IZn+1 )red ). Using the relative Spec, we can generalize this to non-affine case. If X and Z are smooth then we get the total space of the normal bundle. ˜ → A1 which • X can be degenerated into the normal cone, i.e. there is a morphism of varieties X satisfies the following situation: NX (Z) ˜ X X × (A1 \ {0}) {0} A1 A1 \ {0} Compare this with the fact that a filtered space can be degenerated into its associated graded ring: {A locally free coherent sheaf on A1 equivariant with respect to Gm } ↔ {filtered vector spaces}. To describe the equivalence, let E be a locally free coherent sheaf on A1 corresponding to a module M over k[t], and V be a filtered vector space. Then the equivalence is given by E 7→ E1 = M/(t−1)M with the filteration (E1 )i = im(Mi → M/(t − 1)M ) and V = Vj ⊃ · · · ⊃ Vi+1 ⊃ Vi = 0 (i  0, j  0) 7→ M, Mi = V≤i . Theorem 1.1 (Riemann-Hurwitz formula). Let f : X → Y be a morphism of smooth irreducible curves. Then k(X)/k(Y ) is a separable extension. Recall from the lecture on September 22th, that for x ∈ X we have the ramification index d at x if the divisor f −1 (f (x)) has coefficient of the irreducible divisor x equal to d. This is equivalent to saying that in the extension of DVRs Oy,Y ⊂ Ox,X , (valOx,X )|Oy,Y = d · valOy,Y . ∗ Let dx be the ramification index at x. Assume Xthat dx is prime to char(k). Then f KY → KX extends ∗ to an isomorphism f KY (R) ' KX where R = (dx − 1)x. x∈X Corollary 1. If X, Y are complete then deg KX = n · deg KY + X (dx − 1). x∈X Let’s consider the example of elliptic curves. Let X be the projective plane curve defined by the equation y 2 = x3 + ax + b. Then the projection (x : y) 7→ x extends to a map X → P1 , which is ramified at the roots of the polynomial P (x) = x3 + ax + b and the point at infinity ∞, with a unique point over each ramification point. Moreover, from the adjunction formula, deg KP1 = −2. Therefore, deg KX = 2(−2) + 4 = 0. Observe df that if x ∈ X is a smooth point on a curve and f is a function on X not equal to 0 with f (x) = 0, then f has a pole of order exactly 1 at x, i.e. it is a local generator of KX (x) (an exception is when char k = p, (f ) = nx (x) + (other points), p|nx ). If f ∈ mx /m2x then df is a local generator for KX ' ΩX . In general, if f = ϕg n where ϕ(x) 6= 0 and g ∈ mx /m2x , then d · deg f = d · deg ϕ + dnx · deg g. Now, take f ∈ mx ⊂ Ox,X . Then f ∗ KY → KX extends to a local isomorphism f ∗ KY (y) ' KX (x), where f ∗ KY (y) = f ∗ KY ⊗ f ∗ y and X ∗ similarly for KX (x). Therefore, f KY (R) ' KX ( x). dx >1 Recall that a smooth irreducible variety is normal, but the converse is true only in dimension 1. Proposition 1. Let X be a normal irreducible affine variety and X ⊂ X be a closed subvariety. If dim Z ≤ dim X − 2 then k[X] = k[X \ Z]. Therefore, for normal varieties, the regular functions extend from the complement of a codimension ≥ 2 closed subvariety to the whole space. 1 Lecture  18.725 Algebraic Geometry I Proof. We may assume that X is irreducible. Using induction on dim Z we can reduce to showing that any f ∈ k[X \Z] is regular generically on Z, i.e. there exists an open subset U ⊃ X \Z such that f is regular on U . Suppose that this is not true for some f ∈ k[X \ Z]. Then f generates a coherent sheaf F ⊂ Rat(X) where Rat(X) is the sheaf of rational function on X, such that F |X \Z ⊂ O. Thus F /F ∩ O is coherent, supported on Z, and killed by IZm . After modifying the choice of f we can assume that m = 1, i.e. IZ (F /F ∩ O) = 0. Thus, for any ϕ ∈ IZ , ϕf ∈ k[X], but for any open subset U ⊃ X \ Z, f ∈ / k[U ]. Now we claim that for any ϕ ∈ IZ , ϕf ∈ IZ ⊂ k[X]. Indeed, by the hypersurface theorem, ϕ|D = 0 for some Weil divisor D ⊃ Z. Suppose that z ∈ Z and ϕf (z) 6= 0. Then ϕf 6= 0 on some neighborhood U of z and, by assumption on f , f is not regular on D ∩ U , a contradiction. Hence, ϕf ∈ IZ . By replacing ϕ with ϕf , we obtain that ϕf 2 ∈ IZ . Using induction we conclude that ϕf n ∈ IZ . To get a contradiction it is enough to check that {f n } generates a finite OX -module. But, by the previous argument, f n ∈ {ψ |IZ ψ ∈ k[X]} ⊂ (ϕf )−1 k[X], the last one being a finite OX -module. Therefore {f n } generates a finite OX -module, finishing the proof. Note that the normality assumption in the above proposition is necessary: Let A = {a0 + a2 P2 + a3 P3 + · · · }, where Pi is a homogeneous polynomial in n indeterminates of degree i. Then Spec(A) is non-normal with the normalization An → X = Spec(A), which is bijective and an isomorphism away from zero. However, A = k[X] = 6 k[X \ {0}] = k[An \ {0}]. We say a set is constructible if it is a finite union of locally closed subvarieties of Y . Theorem 1.2 (Chevalley’s theorem). Let f : X → Y be a morphism of varieties. Then: • im(f ) is constructible. • Furthermore, if we assume that X, Y are irreducible and that im(f ) is dense in Y , then the function on im(f ) given by f (x) 7→ dim f −1 (f (x)) (the dimension of the fiber) is upper semi-continuous. In other words, for any d, {f (x)| dim f −1 (f (x)) ≥ d} is close in im(f ). • Finally, under the previous assumptions, there exist a non-empty open subset U in Y such that dim f −1 (y) = dim X − dim Y for all y ∈ U . Lemma 1. Let f : X → Y be a morphism of irreducible affine varieties with im(f ) dense in Y . Then there finite,onto π 1 is a nonempty open subset U ∈ Y such that f −1 (U ) → U factors as f : f −1 (U ) −−−−−−→ U × An −→ U. Proof. Let K be the fraction field of k[Y ]. Consider k[X] ⊗k[Y ] K which is finitely generated over K and has no nilpotents. We can apply Noether normalization lemma to find f1 , · · · , fn ∈ k[X] ⊗ K = A such that A is finite over k[f1 , · · · , fn ]. Let {gi } be generators of k[X]. The {gi } must satisfy monic equations over k[f1 , · · · , fn ]. We can now choose U so that all fi and the coefficients of the equations are in k[U ]. The lemma implies that if f has a dense image, then im(f ) contains a dense affine open subset. Proof of Chevalley’s theorem. The first part of the theorem now follows from the implication of the lemma, and by Noetherian induction. To prove the remaining, we can assume, without loss of generality, that X, Y are both affine. By the lemma, obtain an open subset U in Y such that dim f −1 (y) = dim X − dim Y , ∀y ∈ U . Use the hypersurface theorem and induction on dim Y to conclude that the dimension of every nonempty fiber is at least dim X − dim Y . Now, using Noether normalization for Y , obtain a finite surjective morphism g : Y → Am where m = dim Y . Let z ∈ Am and y ∈ im(f ) ∩ g −1 (z). Then the fiber f −1 (y) is a union of components of (gf )−1 (z). By the hypersurface theorem, every such component has dimension ≥ dim X − m. 2