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(1)Remarks on the Geometric Genus of Hypersurface Isolated Singularity. by. Yoshiaki TASHIRO", Takako SUZUKI** and Teiichi HIGUCHI"* --. The theory of toric varieties is a beautiful, powerful subject which is finding. more and more uses. Unfortunately its complexty (i.e., the dual of the dual) makes it rather diflicult to use.. This short paper consists of some instructive elementary examples and applications collected as we were learning the subject.. After reading this, the reader could go on to a detailes introduction such as [Oda], and then tackle Munford's paper. In this paper, we focus on the resolution of hypersurface isolated singularity. by totus embbedings. This resolution can easily be constructed in the (special). cases when the defining equation is non-degenerate with respect to its Newton boundary. We then give an application of the resolution above. The geometric genus of a hypersurface isolated singularity is easily computed via resolution.. The material of this paper was developed in discuss with K. Watanabe. We recall a few preliminaries related to the concept of the geometric genus of normal isolated singularities in g1 and torus embedding in g2.. In g3, we show the figure of the resolution of hypersurface isolated singu-. larities. ･. '. tv. 1. Geometric genus of hypersurface isolated singularities. Let (X, x) be a normal isolated singularity in the n-dimensional analytic space. X. It follows from Hiromaka's work [6] that a resolution n:X-X always exists. '; Definition. The geometric genus of a normal isolated singularity (X; x) is '. '. P,=dimc(Rn-i z*e7)x. ･. * Tokyo University of Agriculture and Technology ** Dept. of Mathematics, Yokohama National University.

(2) 14 Y. TAsHiRo T. SuzuKi and T. HiGucHi The geometric genus is in fact independent of the choice of the resolution. Yau [7] derived an intrinsic definition of p, that does not involve a priori know-. ledge of what a resolution of X looks like, which is a generalization of Laufer's thorem [1] in the 2-dimensional case.. Theorem. (Yau [7]) Let x be a norma! isolated singularity of X. Suppose that V is a (suthciently small) Stein neighborhood of x and Kis the canonical line. bundle of V-{x}.. Then P,==dim r(V-{x}, e(K))fL2(l/-{x}), where L2(V-{x}) denotes the set of all square integrable holonorhic n-forms on V-{x}.. Let (7=z'i(V) and A==n-'(x), then T(U, O(K)=L2(U-A) by [1] (Theorem 3.1.. p.601).Thereforeweobtain ' ' p,-dim r(U-A), o(K))lr(U, O(K)) Lemma. Let f(zo, zi, ･･･, g.) be the defining equation for a hypersurface X with an isolated singularity at x, the origin. Then there exists a nowhere vanish-. ing holomorphic n-form on X-{x}. ProoL On X, f(go, gi, ･･･, a.)=O. So. fz,dgo+fz,dgi+'''+fz.dgn=O･. Then. aziA･･･Adz.dg2A･･･Adg.Adzo dzoA･･･Adg..i. W= fzo == fxl ='''= fZn'. Since x is an isolated singularity, f2,=fz,=･･･=fz.=O occurs only at x. Hence. on X-{x}, to is well defined holomorphic n-form, which is nowhere vanishing. The singularity showed in this Lemma is called a hypersurface isolated singularity.. Generally, n-dimensional hypersurface isolated singularity is denoted as follows.. Assume f is a holomorphic function defined in the neighborhood of'sthe origin in. C"+',thesetX={f(x)==O}hasanonlysingularityattheorigin. . This singularity becomes necessary a normal singularity. From now on, we treat only hypersurface isolated singularities.. Now, for any element 0 in l'(I7-{x}, e(K)), using to in the above Lemma, 01w. is a holomorphic function on V-{x} where V is a Stein neighborhood of x. Let g=0/to. Since g is a weakly holomorphic function on V and x is normal, g becomes a holomophic function on Vl So, if to is a square integrable holomorphic n-form, 0=gto becomes square integrable.. Converse is trivial. That is (X, x): rational singularityOto: .,square integrable i.e. P"g(X, x)=O , In the case P,>O, tu is not square integrable. Following Yau's Theofem [7]',.

(3) RemarksontheGeometricGenusofHypersurfaceIsolatedSingularity 15 we can count geometric genus by investigating how many not-square integrable holomorphic n-forms including te are on V-{x}. Since L2(V-{x})Ei!l-'(U, O(.K)),. gw￠L2(V-{x})Oz"gw￠l"(U, 0(K)) t. The latter case is that T*gtu has poles on A. As g is holomophic also at the. origin x. Let g=ZCa,, xay"g" be the Taylor expansion. If the values of 2, pt, p in z*gto=T*gz*w become enough large, there will be no poles of z*gto on A. ei.. We investigate the extend of the values by using embedding resolution. The result is that. Pg(XL x) =#{(Zo, ･･･, 2.)eArn"1(Zo+1, ･･･, 2.+1)GI"-(f)}. where N is the set of all non-negative integers. L(f)=V{(2o, ''', 2n)1li(2o,''',2n)gO}. i. li: defining equation for a n-dimensional compact face Ai of Newton polyhedron T(f) (i=1, 2, ･･･). 2. Torusembedding. Let f(go, ･･･, 2n) be a germ of an analytic function at the origin such that f(O)=O and f has an isolated critical point at the origin. We assume that f has. a non-degenerate Newton boundary in the sense of Varchenko. 'Let X be a germ of hypersurface {(Z6, ･･･, Z},)cC"'ilf(g)=O}. Let f(zo, ･･･, zn)=￡ a.z" be the Taylor expansion of f where Z"=zo"o･･･ zn"n. Recall that the v boundary T(f) is the union of the conpact faces of T+(f) where r+(f) is Newton. the convex hull of the union of the subset {v+(R.o)n"} for v such that a,¥O. J. For any closed face r of T(f), we associate polynomial f,(Z)=Z a,g". We say vEr that f is non-degeneratle in the sense of Varchenko if f has no critical point in. '. (c*)n+i for any reT(f)･. ' Nowweshowthecasewhenn==1. Let f(x, y)= Z aa,xRy" be the Taylor expansion in the origin (eC2). {(2, pt)+(R.o)2} denote the col!ection of (2, pt)+(R>o)2={(x, y)GR21xl2, yl-ilpt} for 2, pt such t,hat aa, ¥O,. where (R>o)2 is {(x, y)lxlO, ylO} (figure 1) T(f) is the convex the hull getting grfOIPhethbeo?dbOfVaeceOnleifigfiignUrtehe2)fiagnudrer2(,f) Which is compact face of T.(f) is the part.

(4) Y. TAsHIRo T. SuzuKI and T. HiGucHi. 16. (x,u). (x,u (x ,p). (x,v). (x,u). (x,u). o. x. Fig. 1 y. 7I. y. (O,4). r+ (f). (lr2J x. o. O (2,O). x. Fig.2 Fig.3 Ex.ample 1. An example of non-degenerate function. Let f==x2+Rxy2+y`. T(f) is the union of compact faces of the convex hull T+(f). T(f) consist of three parts:. (i)point(O,4) @point(2,O) @segmentfrom(O,4)to(2,O). Let (D be r, f,=y`. aof.;' = lfy'=O, i.e. O=4y3=O. Therefore the solution of the above equations is (a, O) for aeC.. Ofr Ofr. Let @ be r, f,=x2. o. = oy =O, i.e. 2x==O=O. Hence the sglution is (O, P). for PGC. . ･ Let @ be r, f,=x2+2xy2+y`.. Ofr = 2x + 2y2. Ox. Ofr =2Rxy+4y3. From Oy. Ofr o. ==Ofr oy =O, '2y3+4y3=y3(4-Z2). Put y=O, then (O, O) is a solution.. Put 2=±2, then (±1, 1) is also solutions. This means f is degenerate..

(5) RemarksontheGeometricGenusofHypersurfaceIsolatedSingularity 17 When 2= ±2, this is not non-degenerate. Let N' be the space of non-negative vectors in the dual space of R"". For any vector a=(ao, ･･･, a.) of N+, we associate the linear function <a, 2>=Zn ai2i. i==o. on r+(f). And let l(a) be the minimal value of <a, R> on T+(f). and let r(a) = {ReT+(f)l<a, 2> :l(a)}. We introduce an equivalence relation -v on N'f by a-vb if and only if r(a)= r(b)･. For any face r of T+(f), let r*={aeN'lr(a)=r}. The collection of r* gives a rational partial polyhedral decomposition of AX' which we call the dual Newton diagram of f and we denote it by r+*(f). 7(a) is a compact face of r(f) if and only if a is strictly positive.. Now we shows an example of dim n= 1.. y. <a, R>== aoRo+ai2i is a line whose normal vector. iS (ao, ai) for a=(ao, ai)GN" and 2=(2o, Zi)eR2.. //･. '. r. I(a) is the constant term of the equation of the. /. line * in the right figure. r(ao, ai)= {(2o, 2i)GI"+(f)laoRo+ai2i=l(ao, ai)} is. one of faces of r(f), namely the faces of the '. '. ,. '. intersection of the line <a, 2> and r+(f). (to see x '. o. the figure below) Fig.4 I/</ 7//. yy. r+(f). (f). x. o. Fig. 5. x. o. '. Fig. 6. r. <a, 2> is parallel to a face of T(f). /. y. We introduce an equivalence relation tv on N+.. j i. In the right figure, let the normal vectors of. 1. land m be (ao, ai) and (ao', ai') respectively. Then. (ao, ai) and (ao'.ai') become B, too. Therefore (ae, ai)-v(ao', ai'), that is, the lines whose normal. vectors have the slope between the line AB and l. l. x. line BC are equivalent. And for any face reL(f), let. Fig. 7.

(6) Y. TAsHIRo T. SuzuKi and T. HiGucHi. 18. / ". o. 'o '. ×. .. X･, .. ... -s S. Ax. ×. Fig.8 Fig.9 '' dualNewtondiagram Newtonpolypedron, ･, c---"-. Fig. 10 dual figure. r*={(ao, ai)GIV'Ir(ao, ai)=r}･. '. The dual Newton diagram r+*(f) is showed in the following figures. In this case, the number of equivalence class is seven. Referring this figure, it is clear the points in r+(f) correspond to the faces in T+*(f), and faces in I'+(f) correspond to the points in r+*(f), and lines in Ii+(f),. and lines in T+(f) correspond to lines in T+"(f)., ' That is, the dimension of a face in r+(f) plus the dimension of the corresponding face in T+*(f) equals two. s. Now using only points and lines, we simplify the above figure in the right. figure. ･. Next, we draw figures in the case n=2... We consider the Newton polyhedron whose Newton boundary is only a face. v+z/+z=1. , , , .,･,･,･. Dual Newton diagram (This is considered as the solid which is seen from the. pointatinfinity). , ' ･ . Againwedrawthedualdiagramoftheabovedualdiagrarn.- . .,. ThisdiagrambecQmesequivalenttothefirstNewtonpolyhedrQn. '.･.

(7) Remarks on the Geometric Genus of Hypersurface Isolated Singularity. zx- lllane. z. Z-atXIS. yz-plane. (O,O,1). i. (O,1,O). xz-plane z-plane. (1,OtO). (1,1,1). z-axls. ,- :. 19. x+ ;z=1. x+y+z=1. S y-axis. o, 1,O). y. xy-piane. ,. (O,O,1) - xy--plane Fig. 11. (1,O,O). x- Fig. 12. L. (O,1 ,o). (1. o,o) '. {O,O,1) Fig. 13 Next, we make the unimodular subdivision of T+"(f). In C2, the cone which link the origin 2 points (a, b) and (x, y) respectively is. unimodular O the determinant. ab. ==1. This means the area of the parallelogram with the two segments (O, O) (a, b) and (O, O) (x, y) equals 1.. In C3, the segment between (a, b, c) and (x, y, z) is unimodular. o G.c.M ofIZ ,b,￡ 2, C, .al-i. '. The plane determined by the three points (a, b, c), (x, y, z), (u, v, w) is uni-. .a bc ". modularOthe determinant x y z=1. . ･ uvw. This means that the value of the volume of the parallelepiped determined by the vectors (a, b, c), (x, y, g) and (u, v, w) through the origin equals 1. ･/' i.. :.

(8) Y. TAsHIRo T. SuzuKi and T. HIGucHI. 20 t. y. y. 1 1 ' '. 2). 1)･. l. (i,}('. '. o. {oO,1). x. (1,O). l. 1. '. 1. 3,2> ' e.-(2,1). /. z. v o. '. -. '. (lrO). x. Fig. 14. Fig. 15 Example 2. In the left figure for the purpose of unimodular subdivision of the two plane, we can draw the line in the figure 15.. g,o-2 g(6)+g(g)-(?) here. 10 21. 21. =1 3 2' =1 g?==3 g(;)+g(2)==(l). and == 1. 32 11 11 O.1. =1. Let A(f) be a unimodular subdivision of I"+*(f). For each n-simplex '. '. '. ao,o ao,1... ao,n al,o am ... al,n. - - --i- - ---an,o an,1 ''' an,n. we associate an (n+1)-dimensional Euclidian space C"+' with the coordinates (u.,o･････u.,n) and birational mapping T.: Cn+'-C"+i which is defined by gj= (u.,o)"o'd(za.,i)"i'j'･･･(a.,n)an'j'. Then the torus embedding XLt(f) associated with a(f). is the union of C""' which are glued along the image of T.. Let z be the projection. and let X be the closure of n'i(X-{O}). It is known that'n: X-X is a reso-. lutionofXL ' ' ･ '. Now, we consider the case n=2.. Choose one cone a in A(f). Let (u, v, w)EC.3, (x, y, g)EC3, is showed in the. fQllowing figure,. ･j.

(9) RemarksontheGeometricGenusofHypersurfaceIsolatedSingularity 21. (u, v, w) (Z; bbj cCII) y (aj,bj,cj). '. Xaic bic cic1 ' (rbi,ci). .v(uazvaj'waic,ubivbj'wbic,uCivC]'wCk) atb k) =(x, y, z). x. Weprepareq3asthesamenumberascones o revealed b.yunimodular subdivision. For example Fig. 16. we consider the coordinates (u.pt v.j,' w.,.)GCRi which maps to C3 by z.j, and the coordinates (u.i, v.i, w.,)GC.3,, which maps to C3 by T.i. t/.). Assume both (za.j, v.j, w.j) and (u.i, v.,, w.i) map to the same coordinates (V, zl, z)EC3.. (x,y,g) = (uaj･vaj･waj･)Aad = (ntaiVaiWai)Aai. where A.j and A., are (3, 3) matrix corresponding to each cone oj and oi. Since IA.ii=1, there is inverse matrix of A.i. So (x, y, g)A.i-i=(ua,v.iw.i)･ Then (UaiVaiWai)=(Uqj･Vaj･Waj･)Aaj･Aai-'･. That is, this determines the equivalence relation between (u.i, v.i, w.,) and (UadVa]･Waj･).. (Uai, Vai, Wai)nw(Uapt Vapt Waj･)O(Uai., Voi7 Wai)==(Uapt Va.i, WaJ･)Aaj･Aai-i,. This contents also transitive low by the composition of the mappings. This is clearly an equvalence relation.. u C,1･v (where ll means disjoint union) is the torus embedding associated with the unimodular subdivision.. Example 3.. y. y 1 ,. (o ' 1). '. 1. t. t. t. (1 ,1). ' '. '. {o. '. ･x. o. (1 ,o). '. ,1) ' t. o. A (f). '. t(1 ' 1>. '. (1, o). .×. Fig.18 ,. Fig. 17. dual figure of A(f). We. represent C2 as. [IZ , and we prepa're ' two C2s because of the number '. of cones is two.. Tal: Ca2i. -C2. ' ' t t ttt ' (u,v)(l 2)--(uv,v)=(x,y)' ''.' . ..

(10) 22 Y. TAsHiRo T. SuzuKi and T. HiGucHi then. (u, v)=(x, y) (-.I 2)=(xy", y-'). Ta2:Ca22'C2 (p,q)(6l)t-(p,pq)=:(x,y) ' (u, v)--(p, q)o(u, v)==(p, q) (6 l)(-l 2). x. -(p, q) (--2 l) =(q-i, Pq). '. c3, :-:q'--'--})Xxso2 x '. .,,,,. u y vlJ----N;.>x. tt. 1･. c2. a2 ,･1 tt. Fig.19. /. //r'. tt ttt t. This shows u==-:i-. We can see that there is Pt,./embe9, ded in the glued. Example 4. , f==x3+y3+xy. ///. /. O,3. . /t. -,J. ･. (f). (1 (1 ,2>. r(f)+. (o. (1,1 o. tr+. ''. x. t. 2, 1). 1. o. (1 ,o). part.. ,1). x. A(f). '. (1. o. '. (2, 1). (1 ,o). x. Fig. 20 Fig. 21 Fig. 22. Newton polyhedron - dual Newton diagram - unimodular subdivision. since S2-i, .8?=i･ ?5==3 modli/la(r;s)u+btl/v(.si.o)･n=a((i).' We add one pomt (i, i) m the middie' cone. (uni-.

(11) RemarksontheGeometricGenusofHypersurfaceIsolatedSingularity 23. So,wehavetopreparefourC2sfortorusembedding. .,, (u, v) (S 2) -- (uv2, v). we blow up f by this mapping.. (uv2)3+v3+uv2v=v3(uSv3+1+za) whenv=O, 1+en=O (u, v) (? l) t-- (u2v, uv) (u2v)3+(uv)3+u2vuv==u3v2(u3v+v+1) (u･ v) (l S) -s- (uv, uv2). (uv)3+(uv2)3+uvuv2=u2v3(u+uv3+1) ). (u, v) (8 ?) --･ (u, u2v) u3 + (u2v)3 + uu2v = u3(1 + u3v3 + v). For the purpose of getting the figure of the torus embedding, we draw a dual diagram of a(f).. Y uv/JPI -v. Rl V7?1 --+. '. u2 ￠. (r2) '. 1). t '. !). if x ･' ;i 'j ;･. (1,rO). tt-tt tt Fig.23 . Fig･24 ,,,.L-. ,, ,,r 'i･･'. t hoefnffhiS t ?, ree Pi beCOmeS one pomt, this curve becomes 1ike this is the figurW. '. '. Fig. 25. ' tt i( The curve abovg is made untied by the embedded resolusion. We call such a resole,, tion a?..r` embedded resolution by torus embedding ". : !. 3. E'mbedded resolution by torus embedding.. t/1 ... ordeYere, We ShOW hOW tO COMPIete the embedded resolution in the following. 1.

(12) 1. 24 Y. TAsHiRo T. SuzuKi and T. HiGucHi 1 draw Newton polyhedron, 2 then, draw dual Newton diagram r+(f), 3 do the unimodular subdivision of T+*(f),. 4 determine the map z., which associate with each cone appeared in a(f), 5 further, draw the dual diagram of r+*(f), 6 put the coordinate functions dual to the corresponding axis. 7 blow up f by z.i･. Example1 Let f=x2+y2+g2 be the difining equation for a hypersurface singularity. The equation of its Newton boundary is s. ie + 2L + 3L ..1. 22. 2. Z g.g.:.1 (o,1,o) (1,O,O) c. (otO,2) gx'+y+z.2. A (lgl,1) (O,2,O) (2,O,O). y. x (O,O,1) Fig. 26 Fig. 27. Since the normal primitive vector of this plane is (1, 1, 1), the dual Newton. diagram is drawn as above.. We make a unimodular subdivision. Here, all the lines are unimodular. And. 111 111 111 OOI 100 OIO. O1O=1 OO1=1 1OO=1 So all the planes are unimodular, too.. r. (o,i,o} (i,O,O) 111 ) A (u, v, w) (. u. v w. W. (1 1,1 V. wu. v. B (u, v, w)(. C (u, v, w) ( (O.O,1) Fig, 28. o. 1. o. o. o. 1. 1. 1. 1. o. o. 1. 1. o. o. 1. 1. 1. 1. o. o. o. 1. o. O (u, uv, uw) =(x, y, g). ) ). O(uw, u, uv) =(X, y, g). D(uv, uw, u) :(X, Y, 2).

(13) RemarksontheGeometricGenusofHypersurfaceIsolatedSingularity 25 A: u2+(uv)2+(uv)2==u2(1+v2+w2) -This joins v axis at two points and w axis. at two points, too. '. B: (uw)2+u2+(uv)2=U2(w2+1+v2) C: (uv)2+(uw)2+u2=u2(v2+w2+1). Exactly, 1+v2+w2=O is the equation of the exceptional set.. '. '. u. vw ￠3 u. P2. Wv ￠3 ￠3. v wu. Fig. 29 Fig. 30 So the figure of the exceptional set may be considered as the right circle. (bold face line of the figure is the dual of dual Newton diagram) In this figure, the neighborhood of the origin of the coordimates (u, v, w) is. locally homeomorphism to C3, and P2 is embedding in the inner part. Then, the state of the resolution of singularities is considered as the following figure.. z. T. -s Nl. x. y N. s. s. N. Fig. 31. Example 2 Let f=X2+Y3+Z6 be the. 2+. K.. defining equation for a hypersurface. singularity. -. We examine whether the seqments are unimodular or not. O @ @ are evidently unimodular.. @g?=3 ?6--i ' ,2 1 @g3-o ,,-2. '. 21 32 @io,==-2 oo=O. 13 oo 13 10 i3 Ol. =o. O.K.. = -3. O.K.. =1. O.K..

(14) Y.･ TAsHiRoT.SuzuK･r･andT..HiGucHi･,.･ ,･,.'･. 26. f '' (O,1,Oi. '''',..?,'.,,[''l (1,O,O) z. )+ g 3X. (O,O,6). g+g-i ,' ,. ., .. +2y,+z='6i, '',2- M3 '. (O,3,O). 5. '. y. ttt. (2 , orO). (O,Q,1) Fig. 33. x Fig. 32. Hence, all the segments are unimodular. Next, we examine whether2 th2 faCeS are unimodular or nbt.. 3. '. Il. o. o 3. O=1 ･ O.K.. .. l2 1, ･iL:･ '1･1'lliiii. 10=3 NO [i,. II O o. Ofacel II. we have to'seak c,, c2 (1 or 2) of the equation. Making subdivision of the tt '･ ,, g( i)+g'(:)+g'(1)-(Sl). '. where 3 is the determinants of face II and lei, k2, le3 are intbger.. As the equation. g( i)+g( g･ )t g( i)-( l･). satisfy the condition, we puta POtih'net (tih'reie' it)r'iangies iinking' ffbm 'e'ach vertex of. Then we examine Whether the face II to (1, 1, 1) unimodular. Then. 11 1' Ol 1==1 O.K. oo 1 (;2 )Oii;, subdivision is needed. '. S. O,r4gt',' i' 32'i' ' 6i8l=l, ･O･11K･ '2･l 2,-2 (f i,,r). ih;, (al,Si i?cs same as ?- eforei.

(15) Remarks on the Geometric Genus of Hypersurface Isolated Singularity. 27. here we put a point (2, 2, 1) in this face,. t. t:t tttttt. 32'1. 221 O.K , O1O.==1 O.K O1O=1. ttt t. t t t. 221 111 t. tt.. Then we finished the unimodular subdivision of the face. 3. 2. 1. 2. 2. 1. 1. 1'. 1. =1. II.. III 3 2 1. O O 1=2. 100. g(i)+g(e)+g(k)-(i). e.. We put a point (2, 1, 1) in this face, then. 321 211 3 211 100 1. OO1=1 O.K. OO1=1 O.K 2. 2. 1. 1. 1. 1. 1. =1. O. K. Hence all unimodular subdivition is finished.. Then, let the divided faces (unimodular subdivision) be named from A to I, we can draw the next figure.. ' (6,i,o) a,o,o) A. cB D. 3,2,. 2,pu. H (2 ,1' , 1). (lrltl. EF. G. .tT. t. (o,o ,1) Fig. 34. A (.,v･w) (i ;t. 1) '"v (uw3, vw2, w). u =v=O ･･･2 point v= w=O ･･･ 2 points. (uw3)2L+,(vw2)3+w6. .#w6.(u2+v3+1). .. W=U=O.e,''3 pomts.

(16) 28. Y. TAsHiRo T. SuzuKi and T.'HiGucHi. B (u,v,w) (g ? 61A.(u3w2,u2vw2,uw) u==v=o･･･2points. K22il ･ v=wnyo･･･opoint. (uPw2)2+(u2vw2)3+(uw)6=u6w4(1+v3w2+w2) W==U==O'''OPOint. C. (u,v,w)(g?''61.v(u2w,u2vw,uw) uF･v=o･･･opoint. Ki i il v=w=o･･･opQint. (u2w)2+(u2vw)3+(uw)6.,,u4w2(1+u2v3w+u2w4) W=U=O'''OPOint. D (u, v, w) (8 l 61 ･-v (u, uv, zaw) u==v=o･･･opoint. Xo o il v==w=o･･･opoint. u2+(uv)3+(uw)6,,. za2(1+uv3+u4w6) W=U=O'''O POint E (u,v,w)(66ll-v(uw3,uw2,uvw) u=v':o''･o'point. K3 2 il v=w=o･･･ipoint. (uw3)2+(zaw2)3+(zavw)6 .,, u2w6(1+u+u4v6) W== U =O ''' O POint. F. (u,v,w)(ggh.v(u,w,,u,w,uvw) u=v=o...1point. k2 i il v=w=o･･･opoint. (u3w2)2+(u2w)3+(uvw)6=u6w3(w+1+v6w3) W=:U=O'''OPOint. G (u, v, w) (2o lo ll .N, (u2w, u, uv) u==V=O'''OPOint. Ni o o/ v=zv=o･･･opoint. (za2w)2+u3+(uv)6..u3(uw2+1+u3v6) W=U==O'''O POint. H (u,v,w)(Z?ll･v(u3v2w,u2v,uv) u==v==o･･･opoint. Xi o ol v==w=o･･･opoint. (u3v2w)2+(u2v)3+(uv)6.=u6v3(vw2+1+v3) W==U=O'''3 POintS h. I. (u,v,w) (i ? ll.v(u2vw3,u2vw2,uvw) u=v==O'''Opoint. X3 2 il v=w :o･･･opoint. (u2vw3)2+(u2vw2)3+(uvw)6=u4v2w6(1+u2v+u2v4) W=U=O'''OPOint. The form of the resolution of a singularity is showed in the right figure..

(17) Remarks on the Geometric Genus of Hypersurface. 29. Isolated Singularity. 1O O). (o,i,o). A B. W. u. v. H. c D E. FG ,. '. x. (O,O,1) '. Fig. 35 Fig. 36 Example 3 Let f= x`+y`+g`+xyz be the defining function of ahypersurface singularity.. There are three compact face in the Newton polyhedron in this case. As the face I is a plane which through three points (O, O, 4), (4, O, O) and (1, 1, 1), the equation of the face I is. x+ 2y +g= 4. Same as above,. the face II: x+y+2z==4. the face III: 2x+y+g=4. Then, taking three points (1, 2, 1), (1, 1, 2) and (2, 1, 1) we draw the dual figure. The linked with real lines in this figure mean the intersection with the each faces.,. (O,1,O) (1,O,O} N N. N. N.. ..L. N. A. SNNN N NN ..(2r1 B. -- -- .-. (1,2,1)i. t. ls. css. (O,O,4). (4,O,O). s. 1. t'. (O,4,O). s. N. N. (O,O,1) Fig. 37. /. XD (1,1, 2) s. N ,1,1,. u. z/x. N. rl. F --. lt. SE. `. I. G. ,1) . t. Fig. 38. -. -. '. /. z. 7.

(18) 30. Y. TAsHiRo T. SuzuKi and T. HiGucHi. From the former figure, we have that the all segments are unimodular, For the faces,. AIOO OIO. Face. FaceBO1O FaceC. =1 211 =1 121 211 FaceE112 FaceF D121 OOI =1 OO 1=1 100 112. Face. G121 112. Face. '. =4.. 1. 2. 1. o. 1. o. o. o. 1. 2. 1. 1. 1. 1. 2. 1. o. o. This is not unimodular.. =1. =1. ". 211. (i)+t(l' )+t(i)-(l). AS. we. put a point (1, 1,. 11 12 11 therefore,. 1) in G. Then. 1 111 1=1 112=1111 211. =1. 2 211 121. unimodular subdivision of the faces is completed.. ' (O,1,O). (1,O,O). SN.-. ' '. Ci. NN.. h--N N.1 .A --, N-!z B. NNNs.. F/ NG-. IG -. tst--' '. t i 1 1 l ID E 1 1 N t1 1 ' (O,O,1) Fig. 39. A. (u, v, w) G l 1) -- (uw2, vw, w). (uw2)4+(vw)4+w4+uw2vww=w4(u4w4+v4+1+uv). W. u v. v. t. NN. W. N W. u. s. u. w. '. u e-. u. W Wu 1'. v. it. w u. Fig. 40. ･o pomt v= w= O -- ･o point w=u==O･･･ 4 pomts u==v =O. u.

(19) RemarksontheGeometricGenusofHypersurfaceIsolatedSingularity 31. B (u, v, w) (:t i !.) r- (vw2, uv2w, vzg.), vu.r'!l:Oo lil14 pPoOinnttS (vw2)`'+(uv2w)`+(vw)`+vw2uv2wvw==v4v4(w4+u4v4+1+u) W=U==O'''OPOint. tt. C (.,,,.,(g,i'.･l,)...,.,..2,').,･' -ij,-..-zt;1lil,og,ol:,t tt /ny. 1!il,,',1":i','`. L. points (1,,'W'`i"Wli32"!l.Wili i, IWi;`,'"V' ,,w=u=O･･･4 :'ai-g .' ::2 ,p&}'R,t,'. D -･:. (uw)`+(u2w)`+(uvw2)`+uwu2wzavw2=u4w4(1+u4+v4w4+v) W=za=1'''1POint. (i 6o 26)N(zaw,u,u2v) .. E (u,v,w). (uw)4+u4+(u2v)4+awuza2v=za4(w4+1+u4v4+vw). u= v=O ･･･ 4 points v=w==O ･･･ O point. w=u=O･･･O point. F (U･V･W) (i i, i)-(u2vw,uv,uv2). za =v=O ･･･1 point v==w=O ･･･O point (u2vw)4+(uv)4+(uv2)4+u2vwuvuv2=u4v4(u`w4+1+v4+w) w =u=O･･･4 point. G, (u,v,w) (lt it li)tN･(blvw2,u2vw,uvw). (uvw2)`+(u2vw)4+(zavw)4+uvw2u2vwzavw=u4v3w4(vw`+u4v+v+1). za=v=O･･･Opoint v=w=O･･･Opoint w:u=O･･･1point G, ("･ v･ w) (l l, i) "- (uvw, uvw2, u2vw). (zavw)4+(zavw2)4+(bl2vw)4+uvwzavw2bl2vw=za4v3w4(v+vw4+za4v+1) -. G, (u･ "･ w) (i l l,)n- (u2vw, uvw, uvw2). (u2vw)4+(uvw)4+(uvw2)4+u2vwuvwuvw2=za4v3w4(u4v+v+vw4+1). ' The three curve of the right figure show the form of the exceptional set..

(20) Y. TAsHIRo T, SUZUKI and T. HIGucHI. 32. REFERENCE. [3]. LAuFER, H.B.: On rational Singularities, Amer. J, Math., 95 (1972), 597-608, LAuFER, H.B.: On pt for surface singularities, Proc. Sym. Pure Math. Vol. 30, Amer. Math. Soc. (1976), 45-49. OKA, M.: On the resolution of the hypersurface singularities. to be submitted to. [4]. Proc. of the Japan-U.S. Seminar on complex analytic Singularities, Tsukubal Kyoto, 1984 (T. SuwA and P. WAGREicH, eds.), Advanced Studies in Pure Math. 8, Kinokuniya, Tokyo and North-Holand, Amsterdam, New York, Oxford. HiGucHi, T., WATANABE, K., YosHiNAGA, E.: Introduction to the complex analysis of. [1] [2]. , '. several variables. (in Japanese) Morikita Shuppan (1980).. [5]. HiRoNAKA, H.: Bimeromorphic smoothing of complex spaces. In Lecture Notes.. [6]. Havard Univ. 1971. WATANABE, K.: On plurigenera of normal isolated singularities I. Math. Ann., 250. [7]. YAu, S.S.T: Two theorems in higher dimensional singularities Math. Ann., 231. }. (1980), 65-94. (1977), 44-59.. ". -.

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