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A Resolution of Normal Surfaces of Triple Section Type
Introduction. In this note, "a surface" means an algebraic surface
defined over the complex number field. I11 [5, $21, IIorikawa introduced
a metllocl for resolving singularities on. normal surfaces of double
section type in th.e coinp1.etion of the total space of a line bundle over a
surface. This method is sometimes useful not only for studying surfaces
globally ( [GI, [13], [I41 etc.) but also for studying isolated singularities
locally ([I71 etc.).
I11 fact, let (V, P) be a normal 2-dimentional double point. Since
(;he elnbedded dimension of (V, P) is 3 by [I], an analytic equation of
P is given by t2 + f(x, y) = 0 from a suitable change of the coordinate.
Then one can obtain a resolution of P from a resolution of the plane
curve singularity J(x, y) = 0 by this method. The origin of this idea is
classical ([4], [7], [lo] ,etc.), but the essential point of [5, $21 is to obtain
a forinula o r analytic invariants of P at the same time as resolving P
(15, y.50]).
Now in. tl-lis note, we extend this method to ii.orina1 surfaces of triple
section type. ( i\ special. form o:f this is already seen in [2, $31.) T11.e
local version of our rnethod is the following : Let (V, P) be a normal
2-dimentional llypersurface triple point. By a suitable change of the
coordinate, the equatiou of P is gives1 by E3 + g(x,y)E + h(x, y) =
0. Then, fi-om the pair of the resolution of plane curve siilgularities
g(x, y) = 0 and h(z, y) = 0, one call descend the multiplicity of any
infinitely near point of P ; that is, orre call reduce this triple point to
seine isolated double points and siinple codilnensioil 1 double poii1.t~.
(We call them cornpound nodes and compound cusps.)
111 $1, we produce the above process by twisting the line buildles
sirnil.a,r as in [5, 521, and obtain a certain fonnula. In 52, we give some
examples. I11 $3, we show the following : Let (V, P) be a11 isolated
singularity defined by t3 + +(x, y) = 0. Let /I be the Milnor number