Mumford, David

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A REMARK ON THE PAPER OF M. SCHLESSINGER 11 the conference itself, I spoke on a theorem asserting the existence "semi-stable" reductions for a~lalytic families of varieties over a disc, 00th outside the origin. This talk turiled out to be difficult to transcribe o a paper of moderate size and instead will be incorporated into the tes of a seniinar which I am running together with G. Kempf, B. Saint- at, and Tai, which we will priblish in the Springer Lecture Notes. ere I wollld like to add a footnote to Schlessinger's calculations of formations.' He studied the situation: V = complex n + l-di- I vector space; P ( V )  = n-dimensional projective space of t-di- sional subspaces of V ;  Y c P(V) a smooth r-dimensional variety, 1 ; C c V the cone over Y. Let L = ((jy(l)  ASSLIITIC: flo(~(~),~p(r,(li)) -+ H'(Y, 2) is surjective, k 1 nlay also assrime by replacing P ( V )  by a linear space that it is an iso- rphism for k = I ) .  Then he proved : ) There is a n:rturai i~ijection of f~1nclo1.s: Deformations projective Deformations = (of Y i n  P(v)] /aulornorp~lisms + ( of c b) T,' has a natural graded strr~cture that (T,'), z image of Zariski tangent space to I-7 ,  If (T;), = (0) for k # 0 ,  then A is ison~orphic to the fi~nctor of r~natiotis of C ,  i.e., all deformations of C remain conical. If I .  2 2 and L is sufficiently ample on Y ,  then the condition it1 (c) at I would like to show herc is: 114 RICE UNIVERSITY STlJDIFS d') If r = I ,  L is sufficiently ample or1 Y and Y has genus 2 2 and is not hyperelliptic, then again the condition in (c) is satisfied. This gives: Corollary. There exist normal singularities of surfaces with no non- singular deformation ! To prove (d'), we let U = C - (0) and use the exact sequences: Now C* acts in a natural way on both 19, and N,, and if n: V- (0) -t P(V) is the projection, then both n,8, and n,Nc decompose into direct sums of their eigenspaces for the various characters of C*. Moreover, the C* invariant sections are: and o: induces the map a' = yo /.? ( P  = standard map). Thus we get: a' (T,!),, = coker[r(PH, ("(v + 1) Bc V )  f ( Y ,  Ny(v))], then T: = @ :=?m(712)y. We niust compute these groups. The idea is to determine N, explicitly on Y without actually using the embedding of Y defined by L. Consider in fact N;(1) via the dual of a' as a subbundle of 9, O ,  V* hence for every .KG Y:  REMARK ON 'THE PAPER OF SCfILESSINGER 11 5 ~t is easy to see that  ~lndel. these inclusions, if X ' E  C lies over s: space of linear forms I on V 0; (, , ,(I) 69 cJ.,/117, such that I ( x l )  = 0 space of linear fon11s 1011 V' such that I(.\-') = 0 and I = 0 is tangent to Y at x Hut now by assumptiol~: and 1111dcr this isomorphism, the linear forms I srtch that I=0 and is tangent to Y a t  x go over to the sections of [. vanishing at s to 2nd order, i.e. I'[Y, 111.: .  L )  . Now consider A  c Y x Y with p;L(-36) 1 J P.. 1' with p2 ,.rl):I,( -?A)] . The11 it is easily seen that l ~ 2 , , [ p f l d ( - 2 ~ ) ]  is a locally free sheaf on Y and 111111 G 1-( Y, rn.: * I , )  . the two sub-bundles: P ~ , * [ P T L ( - ~ A ) ]  c p,,+[pTL] = r( Y, L) O, fl'y N ; ( I )  c V* z r( Y,  L )  qual. Now assume I. = I ,  so that Y is a curve and O(-2A) is an in- le sheaf on Y x Y. Then by Serre duality for the morphism p,, we identify N,(- I )  as it quotient of V 8, (0, or T(Y ,  L)* @,.f ly:  116 RICE UNIVERSITY STUDIES We want to show that (T,'), = (0)  if v f 0 ,  i.e., is surjective if v # 1 .  I f  degL > 29, then p,,, of the two sheaves in  square brackets is zero, hence by the Leray spectral sequence for p,, the above map is the same as: We treat the surjectivity in three cases: Case I .  v 2 2:  Consider the sheaf cokernel It is a sheaf of 8,,-modules so i t  lies in an exact sequence between 8,r 0,- modules ( ) x ~ ~ - l @ a ~ - t O  O -+ (O(A) @ O,,) @ L" ' ' @ fly -+ K,, -t (P(2A) 0 0, 0 ?I1 111 l,v - I c-' @ (ay)- l .  So if d e g L >  4g-4, H1(K,j = (0) when v 2 2 Case I ! .  v = 0: Consider the Leray spectral sequence for p,, Since we have assumed Y is not hypereliiptic pi.*Oyxy(2A) 2 ~ l , * f l y , y  and b) R3p1,,ejy y(2A) is a locally free sheaf 6" of rank g - 2. Now we have : Note that 8 does not depend on L. So by (b) there is an integer no depend- ing only on Y such that if degL > no,  then (Q,  @ 8) @ L-' has no sections. Case I l l :  v =< - 1: Surjectivity in this case always follows from sur- jectivity when v = 0 .  In fact, if we know that is surjective, I claim T(Y, N y  @ L-') = ( 0 ) )  v 2 2 .  If not, N y  @ L - ~  has REMARK ON TI-IE PAPER OF SCHLESSINGER 117 a non-zero section s. Then for all t G T(Y, L) z V*,  t @ s is a non-zero section of N y  63 LA' . Thus we must get all sections of N ,  @ I,-' in this way. But this means that all these sections are proportional, hence d o  not generate N y  63 I,-' . But since is s~rrjective and V @ P,  is generated by its sections, so is N y  @ L-'. This is a contradiction, so s = 0 .  This cotnpletes the proof of (d'). Finally two remarks: (A) If yo~i  look at the case Y = P i ,  L = (Opl(li), then C = cone over the rational curve of degree n in P" and the sequences we have ~ised enable us to comp~lte T,' easily. In  facl i t  turns out that if li 2 3 ,  dim (7;!)-, = 31i - 4 .  is most reasonable to conjeclure that the versa1 deformation space this C is a non-singular li - I-dimensional space but with a Q-dimen- nal embedded component at the origin if k 2 4.* B) What happens i n  the hyperelliptic case? If, for instance, n: Y-+ P1 ie double covering and I, = n* Opl(k), then C is itself a double covering he rational cone considered i n  (A) which is known to have non-singular for~iiations. Do these lift to  tlefol.n~ations of this C'! NOTES I .  M .  Schlcssinger, "011 Rigid Singularities," in this voluine, pp. 147 102. 2. H. Pinkliam has recently proved that this i \  true I T  X 2 5, but if k-4, theversal defor- mation space has two coniponents, a snlooth 3-dinlensional one and a snlooth l-dimen- sional one crossing normally at  the origin! (Cf. "Deforn~ations of cones with negative grading," J,  of Algcbt.a, to appear.) 

A Remark on the Paper of M. Schlessinger

https://scholarship.rice.edu/bitstream/1911/63105/1/article_RIP591_part7.pdf

A Remark on the Paper of M. Schlessinger

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