ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 5: SINGULARITIES.
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1 ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 5: SINGULARITIES. ANDREW SALCH 1. The Jacobia criterio for osigularity. You have probably oticed by ow that some poits o varieties are smooth i a sese somethig like that of a maifold from differetial geometry, while some poits are defiitely ot; for example, V (y 2 x 3 )(C) A 2 C has a kik or a corer (the right word is actually cusp ) at (0, 0), but the it of the taget lies of poits o the curve V (y 2 x 3 )(C) as you approach (0, 0) does ot deped o what directio alog the curve you approach (0, 0) from. O the other had, V (y 2 x 3 +x 2 )(C) A 2 C has a loop (the right word is actually ode ) at (0, 0), where the curve itersects itself, with two differet taget lies. Both this cusp ad this ode are sigular poits, which we ow defie: Defiitio 1.1. Let X A k be a variety of dimesio d, ad let x X. Choose a set of geerators ( f 1,, f m ) for the ideal I(X) k[x 1,, x ]. We say that X is osigular at x if the rak of the Jacobia matrix δ f 1 δ f 1 δx 1 δx 2 δ f 1 δx δx 1 δx 2 δx δx 1 δx 2 is d. If X is ot osigular a poit x X, the we say that x is a sigular poit of X. If X is osigular at every poit x X, the we just say that X is osigular. Nosigular varieties bear eough similarities to smooth maifolds that may of the techiques ad results from differetial topology which are used i the classificatio of smooth maifolds ca be imported (with some effort) ito algebraic geometry, where they apply to (at least some) osigular varieties (but usually ot sigular varieties, which are more like maifolds with sigularities). Observatio 1.2. If X is a affie variety, the set of sigular poits of X is give by the vaishig set of the determiats of various miors of the Jacobia matrix. Those determiats are all polyomials, so the set of sigular poits of X is the vaishig locus of some fiite set of polyomials, so the set of sigular poits of X is a closed subset of X. δx Date: Jauary
2 2 ANDREW SALCH Fidig sigular ad osigular poits o varieties is very computatioally approachable: Example 1.3. Suppose k is algebraically closed. I claim that the variety V (x 3 +y 3 xy+1)(k) A 2 k is osigular. The Jacobia matrix is [ 3x 2 y ] 3y 2 x ad (x, y) V (x 3 +y 3 xy+1)(k) is a sigular poit if ad oly if the Jacobia matrix is equal to [0 0] at (x, y). Suppose the characteristic of k is ot 3. Solvig the equatios 3x 2 y = 0 3y 2 x = 0, we get four solutios (0, 0), ( 1 3, 1 3 ), (ζ 1 3 3, 1 ζ2 3 3 ), 1 (ζ2 3 3, ζ ), with ζ 3 a primitive third root of uity i k. Now the questio is: are ay of these four poits i A 2 k actually i the variety? Pluggig each of them ito the equatio x3 + y 3 xy + 1 = 0, we get that oe of these four poits solve the equatio. Hece oe of the poits o the variety are sigular! (The same coclusio holds if the characteristic of k is 3, sice i that case, the Jacobia matrix becomes just [ y x], which is clearly zero oly at (0, 0), ad clearly (0, 0) is ot o the variety.) Exercise 1.4. (Nosigularity i a family of surfaces.) Let k be a algebraically closed field. (1) Let α, β k, ad suppose β 0. We write X α,β for the variety V (x 2 +y 2 +z 2 +αxyz+β)(k) A 3 k. Fid ecessary ad sufficiet coditios o α ad β for X α,β to be osigular. (2) Now let β 0, ad cosider the variety Y β = V (x 2 +y 2 +z 2 +wxyz+β)(k) A 4 k, i.e., we o loger thik of α as fixed, but we treat it as a variable like x ad y ad z! Each variety X α,β is a surface obtaied from the 3-dimesioal variety Y β by itersectig Y β with the hyperplae w = α. Show that Y β is osigular. Exercise 1.4 makes the poit that a variety ca be osigular (like Y β ) but still have closed subvarieties of smaller dimesio iside of it which do have sigularities (like some of the surfaces X α,β Y β ). 2. The local criterio for osigularity. Defiitio 2.1. Recall that, for a affie variety X, there is a oe-to-oe correspodece betwee poits of X ad maximal ideals i O(X). So, for each poit x X, we have a commutative local rig O(X) mx. We call the local rig O(X) mx the rig of germs of X at x. Now thik about smooth maifolds, from differetial topology, agai. A smooth maifold has the property that its dimesio is equal to the vector-space dimesio of its taget space at every poit. Oe could ask whether the Jacobia criterio for osigularity, our Defiitio 1.1, is equivalet to some local criterio for smoothess like that property of smooth maifolds. Ideed it is! It turs out that osigularity of X at x is equivalet to a certai algebraic property of the rig of germs of X at x:
3 ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 5: SINGULARITIES. 3 Defiitio 2.2. Let A be a Noetheria local commutative rig with maximal ideal m ad residue field k. We say that A is regular if the Krull dimesio of A is equal to the k-vector space dimesio of m/m 2. Theorem 2.3. Let k be a algebraically closed field ad let X A k be a affie variety. Let x X. The X is osigular at x if ad oly if the rig of germs O(X) mx is regular. Proof. Suppose X is osigular at the poit x = (a 1,, a ) X A k. Let a X = (x 1 a 1,, x a ), ad we have a k-liear map θ : k[x 1,, x ] k ( δ f θ( f ) = (x),, δ f ) (x), δx 1 δx i.e., take all the partial derivatives, ad evaluate them at the poit x. Now θ(x i α i ) is the vector (0, 0,, 0, 1, 0,, 0), with 1 i the ith coordiate, so θ(x α ),, θ(x α ) is a k-liear basis for k. Furthermore, δ δx h (x i α i )(x j α j ) = 0 if h i, j x i α i if h i, h = j x j α j if h = i, h j 2(x i α i ) if h = i = j so ( δ δx h (x i α i )(x j α j ) ) (x) = 0 i all four cases. Hece θ vaishes o a x. Hece θ ax yields a k-liear isomorphism a x /a 2 x k. Now the rak of the Jacobia matrix of X is the dimesio of the image of I(X) k[x 1,, x ] uder the map θ, as a sub-k-vector-space of k, i.e., it is the dimesio of (a 2 x + I(X))/a2 x as a sub-k-vector-space of a x /a2 x. But, regardig m x as the maximal ideal of O(X) mx, we have m x = a x /(a x + I(X)). Hece dim k m x + rak of Jacobia = dim k mx /m2 x + dim k(a 2 x + I(X))/a2 x = dim k a x /(a 2 x + I(X)) + dim k(a 2 x + I(X))/a2 x = dim k a x /a 2 x =. Hece the rak of the Jacobia is equal to d, where d = dim(x), if ad oly if dim(x) = dim k m x, i.e., if ad oly if the local rig O(X) m x is regular. The local criterio for osigularity is itrisic (i.e., does ot deped o a choice of embeddig ito A k, or ito P k for that matter) ad makes perfect sese for quasi-projective ad quasi-affie varieties, ot just affie varieties, so we use it to defie: Defiitio 2.4. Let k be a algebraically closed field, let X be a variety over k, ad let x X be a poit. We say that X is osigular at x if the local rig O(X) mx is regular, ad we say that X is sigular at x if X is ot osigular at x. We say that X is osigular if X is osigular at x for every x X. The vector space m x really is the right otio of a taget space to X at x: elemets of m x are taget directios you ca move i, alog X, from the poit x. This is most trasparet i the special case X = A k ad x = (0, 0,, 0): the m x = (x 1, x 2,, x ), ad the k-vector space m x has k-liear basis x 1,, x, i.e., the directios you ca move
4 4 ANDREW SALCH i, iside A k, from the origi. But the same idea also works for ay poit o a variety. If X is osigular, the you ca take the k-liear dual of m x ad thik of it as a kid of space of differetial 1-forms o X, but there s a eve better way to set up the differetial forms o a osigular variety (the Kähler forms) ad build up a kid of vector calculus, de Rham cohomology, etc., o varieties; we will retur to this later o i the semester. 3. Completig the local rigs. Recall from commutative algebra that, if A is a commutative rig ad I is a ideal i A, the the I-adic completio of A is defied as the it  I = A/I A. Defiitio 3.1. Let X be a variety, ad let x be a poit i X. By the completed rig of germs of X at x, writte Ô(X) mx, we mea the m x -adic completio O(X) mx /m x of the rig of germs O(X) mx. If X, Y are varieties ad x X ad y Y, we say that x ad y are aalytically isomorphic if the completed rigs of germs are isomorphic as k-algebras: Ô(X) mx Ô(Y) my. Here is oe of my favorite theorems i pure algebra, which I will ot offer you a proof of, but we will make use of it: Theorem 3.2. (Cohe structure theorem.) Let k be a field, let be a oegative iteger, ad let A be a complete regular local rig of Krull dimesio. Suppose that A cotais a field as a subrig. The there exists a isomorphism of k-algebras A k[[x 1,, x ]]. Corollary 3.3. If X, Y are varieties of the same dimesio ad x X ad y Y are both regular poits, the x ad y are aalytically isomorphic. This is a strikig result: after completio, all regular poits o varieties of the same dimesio look the same. Oe cosequece is that, if you wat to uderstad ad classify some family of differet types of geometric sigularities of varieties, your first task is to classify their aalytic isomorphism types, i.e., to classify the complete local rigs at those sigular poits! Example 3.4. (Hartshore s Example ) Let k be algebraically closed. The variety V (y 2 x 3 x 2 )(k) has a sigularity at (0, 0), sice the Jacobia vaishes etirely there, ad (0, 0) is o this curve. The completed local rig at this poit is k[x, y]/(y 2 x 3 x 2 ). Cosult Hartshore for a argumet (it s log eough that I do t wat to reproduce it here!) that this rig is isomorphic to the completed local rig k[s, t]/(st). The crucial poit i this argumet is that, to write dow a isomorphism k[s, t]/(st) k[x, y]/(y 2 x 3 x 2 ), you eed to be able to sed the geerators s ad t i the domai to power series i the codomai, ot just polyomials: you ca t produce the desired isomorphism if you eed all but fiitely may coefficiets to be zero! So this isomorphism of local rigs exists oly after completio. The geometric meaig of this isomorphism is that the sigularity at (0, 0) o the curve y 2 = x 3 + x 2 is aalytically isomorphic to two lies crossig, i.e., the sigularity at (0, 0) i
5 ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 5: SINGULARITIES. 5 the algebraic set V (st) (k) A 2 k (i.e., two lies i the affie plae which cross orthogoally at (0, 0)). Note that eve though V (y 2 x 3 x 2 )(k) is a variety, i.e., k[x, y]/(y 2 x 3 x 2 ) has o zero divisors, the completio of the rig of germs of V (y 2 x 3 x 2 )(k) at (0, 0) does have zero divisors! This ca happe at sigular poits. Example 3.5. (Two differet aalytic isomorphism types of curve sigularities.) Let k be algebraically closed. The variety X = V (y 2 x 3 )(k) has a sigularity at (0, 0), sice the Jacobia vaishes etirely there, ad (0, 0) is o this curve. The completed local rig at this poit is k[x, y]/(y 2 x 3 ). Notice what happes here: the maximal ideal i the rig of germs is (x, y), ad (x, y) 2 = (x 2, xy, y 2 ), but y 2 = x 3 (x, y) 3. This is importat: it meas that, writig m (0,0) for the maximal ideal i the rig of germs O(X) m(0,0), the quotiet m (0,0) /m (0,0) 3, has three k-liearly idepedet elemets (amely, y, x 2, xy) whose squares are zero. Compare this to the situatio (from Example 3.4) i k[s, t]/st completed at (0, 0): the maximal ideal (s, t) = m i the local rig (k[s, t]/st) (s,t) has the property that m/m 3 has oly two k-liearly idepedet elemets (amely, s 2, t 2 ) whose squares are zero. The dimesio of the space of square-zero elemets i m/m 3 is a isomorphism ivariat of a local rig, so this tells us that there caot possibly be a isomorphism betwee the local rigs O(X) m(0,0) ad (k[s, t]/st) (s,t). Eve better, it says that there caot possibly be a isomorphism betwee the completed local rigs Ô(X) m(0,0) ad (k[s, t]/st) (s,t) /(s, t)! Eve though it is possible i geeral for two o-isomorphic local rigs to become isomorphic after completio, i this case, it does t happe, sice m-adic completio (of Noetheria local rigs) does ot chage the fiite quotiets m i /m j.
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