C OMPOSITIO M ATHEMATICA
J OEL R OBERTS
R AHIM Z AARE -N AHANDI
Transversality of generic projections and seminormality of the image hypersurfaces
Compositio Mathematica, tome 52, n
o2 (1984), p. 211-220
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211
TRANSVERSALITY OF GENERIC PROJECTIONS AND SEMINORMALITY OF THE IMAGE HYPERSURFACES
Joel Roberts * and Rahim Zaare-Nahandi **
Compositio Mathematica 52 (1984) 211-220
© 1984 Martinus Nijhoff Publishers, The Hague. Printed in The Netherlands
1. Introduction
Let X be a
variety
defined over analgebraically
closedfield,
k. We will say that X is seminormal ifOX,x
is seminormal in itsintegral
closure for every x E X. For the definition and basicproperties
of seminormalrings
we will refer to
[14]
and[5]
but it seems worthwhile to mention one characterization of seminormal varieties which follows from[14, 1.1].
Saying
that X is seminormal isequivalent
tosaying
that if v = g 0f is
afactorization of the normalization map v:
~ X
such thatf
and g are finite and birational but g: Y - X is not anisomorphism,
then either g is notbijective (as
a map oftopological spaces)
or thereexists y E
Y suchthat the extension of residue fields
k(x) ~ k(y),
wherex = g(y),
isnontrivial. The
plane
curvey2 = x3
is thesimplest example
of avariety
which is not seminormal.
Bombieri
[2],
and Andreotti and Holm[1,
p.91],
have asked whether aprojective variety
which is theimage
of anonsingular variety,
under ageneric projection,
is seminormal. Greco and Traverso[5,
Theorem3.7]
proved
that if X cPg
is anonsingular
r-dimensionalprojective variety
and 11":
X ~ Pr+ 1 is
ageneric projection,
then X’ =03C0(X)
is seminormal.They
alsoproved [5,
Theorem3.5]
that if X is aprojective variety
over anarbitrary algebraically
closed fieldk,
then X isbirationally equivalent
toa seminormal
hypersurface.
In theproof
of thisresult, they
studiedgeneric projections
of a suitableprojective embedding
of the normaliza- tion of X. Butthey
observed that it was not known whether ageneric projection
of anarbitrary embedding
of X wouldyield
a seminormalhypersurface. (Of
course, this was an openquestion only
whenchar( k )
>0.)
Our first main result answers thatquestion.
* Partially supported by NSF Grant MCS-80-01869.
** The paper was written while this author was a graduate student at the University of
Minnesota.
THEOREM 1.1: Let X ~ Pn be a normal
projective variety, defined
over analgebraically
closedfield k,
and let r =dim(X). If
03C0: X ~ Pr+ 1 is ageneric projection
and X’ =03C0(X),
then 03C0: X ~ X’ isfinite
andbirational,
and X’ is a seminormal
hypersurface.
The
proof
of this result uses somegeometric properties
of thesingular
locus of X’.
Thus,
if Y is ahypersurface in Pr+1
1 and x is a closedpoint
of
Y,
we say that Y isanalytically bihyperplanar
at x if(9 Y",
k[[x1, ... ,xr+1]]/(x1x2).
THEOREM 1.2: Let X ~ Pn be as in Theorem
1.1,
and let 7r: X ~ X’ =03C0(X) ~ Pr+1 be a generic projection.
ThenSing(X’) = 03C0(Sing X)
UV,
where v ispurely of
dimension r -1,
and X’ isanalytically bihyperplanar
atevery closed
point of
some dense open subsetof
V.We will prove Theorem 1.2 in Section
3,
as afairly
direct consequence ofProposition
3.1.Here,
we will use Theorem 1.2 to prove Theorem 1.1.The main ideas of our
proof
areexactly
the same as in theproofs
ofTheorems 3.5 and 3.7 of
[5].
PROOF oF THEOREM 1.1:
By [5, Corollary
2.7(vi)],
ahypersurface
X’
~ Pr+1,
or moregenerally
avariety
X’ whose localrings satisfy
Serre’scriterion
S2 [9, §17.I],
is seminormal if andonly
ifOX’,03BE
is seminormal for everypoint 03BE
of codimension 1. SinceSing(X)
hasdimension r - 2,
it will suffice to checkseminormality
in the casewhere e
is thegeneric point
of a
component
of V. Since a localization of a seminormalring
isseminormal
[5, Corollary 2.2],
it isenough
to show thatOX’,x’
is seminor-mal
for some closedpoint x’
of each irreduciblecomponent
of V. NowC2x’,x’
~k[[x1,..., xr+1]]/(x1x2)
for every x’ in some dense open subset ofV,
and it is well known that this latterring
is seminormal. Butseminormality
ofX’,x’ implies seminormality
of(9x,@x, [5, Corollary 1.8],
so this
completes
theproof.
In Section 2 we will prove some results about embedded
projective varieties,
which are needed in theproofs
of Theorem 1.2 andProposition
3.1. Those two results are
proved
in Section 3.We will say that a
ring
A is seminormal if it is a Moriring (i.e.
A isreduced and the
integral
closure isfinitely generated
as anA-module),
and coincides with
+ A,
the seminormalization of A in itsintegral
closure. We will say that a scheme is seminormal if all of its local
rings
areseminormal. In Section
4,
we consider ahypersurface
X’~ Pr+1
1 whosenormalization is a
nonsingular variety.
Let 03C0: X ~ X’ be the normaliza- tion map and let D c X’ be the doublelocus,
defined as in[13].
Proposition
4.1 says, among otherthings,
thatif 0394 = 03C0-1(D)
is a semi-normal
scheme,
then so is D. When X’ is asurface,
thisimplies
that thesingular points
of D must be either nodes ortriple points
withindepen-
213
dent
tangent
lines.(See Corollary 4.2.)
We conclude Section 4by giving
an
example
which shows that bothpossibilities
canactually
occur.A
proof
of Theorem 3.1appeared
in[11],
in aslightly
differentform,
but the connection with
seminormality
was not evensuspected
at thetime. Theorem 1.1 was
proved
in[15] by
the same methods usedhere,
butit was assumed there that X is
nonsingular.
A version of the results of Section 4 alsoappeared
there. Other main results of[15] pertain
toseminormality
forstrongly generic projections
7T X ~ X’ =03C0(X) ~ Pm,
where m >(3 dim(X) - 4)/2
and X isnonsingular.
Those results will bepublished
as a revised version of[15].
2.
Strange
varietiesIf t
0,
then an embeddedvariety X c Pg
is said to be t-strange if thereis a t-dimensional linear
subspace
L cpn
with L~ tX,x
for every closedpoint x E
X.(Here,
and in the rest of this paper,t x, x
denotes the embeddedtangent space.)
Ifchar(k)
=0,
theonly t-strange
varieties arecones. But if
char(k)
>0,
it is easy to constructexamples
oft-strange
varieties which are not cones, for anyt dim(X).
Avariety
which is0-strange
is often said to be strange.LEMMA 2.1: Let X be a closed
subvariety of p n,
and set r =dim(X).
(a) If X is nonsingular
in codimension1,
then X is not( r - 1)-strange.
(b) If X is
not( r - 1)-strange
and X is not ahypersurface
in somePr+1 ~ Pn,
thendim(tX,x ~ tx,y)
r - 1for
all( x, y ) in
some dense opensubset
of
X X X - A.( Here, 0
is thediagonal.)
(c) If X is
not( r - 1)-strange,
then the line xy meets Xonly
at x and y(and is
not tangent toX) for
all(x, y)
in some dense open subsetof
xx x- A.
PROOF: For a
proof
of(c),
we refer to[7,
Theorem2.5]
or[8,
Lemma15].
To prove
(a)
we observe that if Y = X ~L,
whereL c Pg
is an( n - r
+l)-subspace
ingeneral position,
then Y is anonsingular
curve andY ~ P2.
If X were(r - 1)-strange,
then Y would be astrange
curve. But theonly nonsingular strange
curves arestraight
linespl
cPg
and also(nonsingular) plane
conics ifchar(k) =
2.(See [6, Chapter IV,
Theorem3.9].)
Both areimpossible
sinceY ~ P2.
To prove
(b)
we fix 2nonsingular
closedpoints
xl, x2 E X such that L =tX,x1 ~ tX,x2
is(r - l)-dimensional.
Then M =Span(t X,Xl’ t X,X2)
is(r + l)-dimensional.
If the conclusion of(b)
werefalse,
then for everyx ~ Reg(X) = f nonsingular points
ofX}
theinequalities dim( tX,x
~tX,xl)
r -1,
i =1, 2, imply
that either(1) tX,x ~
L or(2) tX,x
c M. Butthe set of
points satisfying
each condition is closed.Irreducibility
ofReg(X) implies
that X must be(r - 1)-strange
or else thatReg(X)
cM,
which would
given X ~ M = Pr+1.
This would contradict thehypothesis.
Q.E.D.
3. Incidence conditions and
generic projections
Consider a
variety
Y cPmk
and a closedpoint x E Y,
and let r =dim( Y ).
We will say that Y is
transversally biplanar
at x( in
the weaksense)
if thetangent
cone C =Spec(gr(OY,x))
is the union of two r-dimensional linear spaces, C =LI U L2,
withdim( L n L2 )
= 2 r - m. We will say that Y istransversally biplanar
at x in the strong( or analytic)
senseif,
inaddition, Y,x ~ C,x.
IfM,
andM2
are linearsubspaces of P n,
thenSpan( Ml, M2)
is defined to be the smallest linear
subspace containing
both of them.PROPOSITION 3.1: Let X be an irreducible closed
subvariety of pn;
assumethat X is not
(r - 1)-strange,
where r =dim(X). Suppose
thatSpan(tX,x, tX,y) is
d-dimensionalfor (x, y)
in a dense open subsetof
X X - A.
If r + 1 m d - 1,
03C0: X ~ Pm is ageneric projection,
andX’ =
03C0(X)
cPu,
thenSing(X’) = 03C0(Sing X) U V,
where :(i)
V is closed in X’ andof pure
dimension 2 r - m;(ii)
X’ istransversally biplanar
in theanalytic
sense at every closedpoint of
some dense open subsetof
V.REMARK: Since
d r + 2,
theassumption
that X is not(r - l)-strange
follows from the other
hypotheses, by
Lemma 2.1. It is alsointeresting
tonote that if
char( k )
=0,
then d is the dimension ofSec(X),
thevariety
ofsecant lines of X.
(See [4,
Lemma2.1]
or[16]). Thus,
ifchar(k) = 0,
it follows thatSing(X’) = qr(Sing X)
for m d. Inparticular,
ifchar( k )
= 0and X is
nonsingular,
then X’ is eithernonsingular
or elsetransversally biplanar
at everypoint
of some dense open subset ofSing( X’).
Before
proving
theproposition,
we will use it to prove Theorem 1.2.PROOF OF THEOREM 1.2: A normal
variety
isnonsingular
in codimension1,
so Lemma 2.1implies
that X is not(r - 1)-strange. By
the sameLemma,
we also conclude thatSpan(tX,x, tX,y)
hasdimension r
+ 2 for(x, y)
in a dense open subset ofReg(X)
XReg(X) - A,
whereReg(X)
isthe set of
nonsingular points
of X and A is thediagonal. Taking
m = r + 1 in
Proposition 3.1,
weimmediately
deduce the conclusions of Theorem 1.2.PROOF OF PROPOSITION 3.1: There is a dense open subset U c
Reg(X)
XReg(X) -,à
such that if( x, y )
EU,
then(a) Span( tX,x, tX,y)
isd-dimensional;
(b)
xy ~ X ={x, y 1,
and xy is nottangent
to X.Here,
xy is theline j oining x
and y. We setY = X X - (0394 ~ U).
Thisnotation will be used
throughout
theproof.
It is known that the set of
( n - m - l)-subspaces L ~ Pn,
with L n X215
= Ø,
is open in the Grassmannvariety
G =G(n, n - m - 1).
It is alsofairly
well known that each of thefollowing
additionalproperties Po, ... , P4
of theprojection
qr = 11" L: X ~ Pm holds for all L in some dense open subset of G.( Po )
7r, : X ~ X’ =03C0( X)
c P "’ is birational.( Pl ) D7T
=((x, y ) E
X X -0394|03C0(x) = 03C0(y)}
has pure dimension2r - m,
or isempty.
(P2) D§ = {(x, y) E Y|03C0(x) = i7(y»
has dimension 2r - m.(P3) S03C0
={x ~ Reg(X)|L ~ tX,x ~ Ø}
has pure dimension 2 r -m - 1,
or is
empty.
(P4)
IfT7T
consists of all non-collineartriples (x, y, z)
such that03C0(x) = 03C0(y) = 03C0(z),
thenT7T
has pure dimension 3r -2 m,
or isempty.
For
( Po ),
see[12, Proposition 3].
For( Pl )
and( P2 )
we consider the closed subsetsZ,
Z’ of(X X - 0394 )
XG,
where Z= {(x,
y,L)|L n xy =1=
,01,
and Z’ = Z ~( Y
XG).
The methods of[12]
can be used to show thatZ has pure dimension =
2 r - m + dim(G),
whiledim(Z’) 2r - m + dim(G).
So if q: Z ~ G andq’ :
Z’ - G are inducedby
theprojection X X G ~ G,
thendim q-1(03BB) = 2r - m (or q-1(03BB) = Ø)
anddim
q’-1(03BB)
2 r - m for all À in a dense open subset ofG,
which is what we need for( Pl )
and( P2 ).
For(P3)
we consider thegeneral
fiber ofthe
projection 03A3 ~ G, where 03A3 = {(x, L )
cReg(X)
XG|L ~ tX,x = Ø}.
Inparticular,
the methods of[12] imply
that E is a closed subset ofReg(X)
XG,
and thatdim(03A3)
= 2r - m - 1 +dim(G). Finally,
ifU3
con-sists of all
(x,
y,z) ~ X
X X X X such that x, y, and z are notcollinear,
and
Z3 = {((x, y, z), L) ~ U3 G|dim(L ~ Span(x, y, z)) 1}
then one can use the methods of
[12]
to show thatZ3
is closed inU3
X Gand of pure dimension
= 3r - 2m + dim(G).
So westudy
thegeneral
fiber of the
projection Z3 ~
G tocomplete
the verification.Thus,
it follows that if L is chosen from a dense open subset ofG,
then 03C0L: X ~ X’ ~ Pm is finite andbirational,
thatD7T
has pure dimen-sion 2 r - m
(or
isempty),
and thatS03C0, T7T
andD’03C0
havestrictly
smallerdimension. Let
M2
=M2,03C0 = p1(D03C0), M2 = p1(D03C0),
andM3 = p1(T03C0)
wherepl is the
projection
of X XX,
or X X X XX,
to the first factor. It is clear that x EM2
if andonly
if thereexists y E X, with y ~ x
and03C0(y) = 03C0(x),
and that x E
M3
if andonly
if there exist y, z E X such that x, y, and z are not collinear but03C0(x) = 03C0(y) = 03C0(z).
ThereforeM3 c M2 . Similarly M2
cM2.
We can also show thatS03C0
cM2,
the closure ofM2
in X. Infact,
if Z is defined as before and
"cl( )"
denotes closure in X X X XG,
then(x,
x,L) E cl(Z)
if andonly
if L ~tX,x =1= j), by [12, Proposition 5].
Thisgives 03B4 (S03C0) {L} ~ q-1({L}),
where 8sends x ~ (x, x ) and q: cl(Z) ~
G extends
q:
Z - G.By considering dimensions,
we find03B4(S03C0)
cD03C0,
sothat
S03C0
cM2,7T’ Therefore,
if L is chosen asabove,
we conclude thatSing(X’) = 03C0(Sing X) U V,
where V =11"( M2),
and that:(i)
V ~ X’ is a closed subset of pure dimension 2 r - m;(ii)
for each closedpoint
x’ in some dense open subset ofV, 03C0-1(x’)
consists of
exactly
2 closedpoints
ofX,
and both of thecorresponding analytic
branches of X’ at x’ aresimple.
(Recall
that theanalytic
branchesof
X’ at x’ are the minimalprime ideals S
cX’,x’
and that if03C0-1(x’)
consists of normalpoints
ofX,
thenthere is a
bijection 03C0-1(x’) ~ {analytic
branches of X’ atx’l
underwhich
x ~ 03C0-1(x’) corresponds
to the idealKer(X’,x’ ~ X,x);
see[12, §3]
or[10,
pp.394-5].
We say that .p issimple
whenX’/x’/P
is aregular
localring.
We knowthat )3
issimple
if andonly
if L nt x,x = Ø, by [12, Proposition 3].)
To
complete
theproof,
we will show that if L is chosen from a(possibly smaller)
dense open subset of G =G(n, n -
m -1),
then X’ =03C0(X)
will betransversally biplanar
in theanalytic
sense at every closedpoint x’
of some dense open subset of V. If03C0-1L(x’) = {x1, x2},
then thiscondition will be satisfied if:
(a)
x,and x2
arenonsingular points
ofX, (b)
L ntX,xl =j)
for i =1, 2,
and(c) Span(03C0L(tX,x1), 03C0L(tX,x2)) = Pm.
(For (a)
and(b),
see[12, Proposition
3 and Lemma3].
For(c)
oneconsiders the natural
homomorphismy j# : Pm,x’
~X’,x’ coming from j :
X’ ~Pm. Then
(c)
says that(j#)-1(p1)
and(j#)-1(p2)
aregenerated by disjoint
subsets of a system ofparameters).
It is also easy to check thegeometric
statement:We recall that
Z ~ (X X - 0394 ) G
consists of all(x,
y,L )
such thatL ~ xy ~ Ø,
and we setZ0 = Z ~ (U G)
where U is the open setmentioned at the
beginning
of theproof.
LetZ,
consist of all(x, y, L ) E Zo
such thatdim(L ~ Mxy) > d - m - 1,
whereMxy = Span(tX,x, lx,y).
Lemma 8 of
[12] implies
thatZ,
is a closed subset ofZo. Moreover, if p:
Z0 ~ U sends (x, y, L) - ( x, y ), then p-1(x, y) ~ Z1 ~ p-1(x, y)
for each(x, y) ~
U. To seethis,
observe that if(x, y) ~
U then there exists an( n - m - l)-space
L with L ~ xy= Ø
anddim( L
~Mxy)
= d - m -1,
because
m + 1 d.
For such an L we have(x, y, L) ~ p-1(x, y) - Z1.
We can also observe that
p-1(x, y)
is irreducible and of dimension =dim(G) - m. (In fact, p-1(x, y)
isisomorphic
to the Schubertvariety S={L|L ~ xy ~ Ø}.) Thus, dim(p-1(x, y) ~ Z1) dim(p-1(x, y))
forall
( x, y ),
and we conclude thatdim(Z1) dim(Z) = dim(G) + 2r - m.
Therefore,
thefollowing property
holds for all L in some dense open subset of G.(PS) D"03C0 = {(x, y) ~ U|03C0(x) = 03C0(y)
anddim(L ~ Mxy) > d - m - 1}
has dimension 2 r - m.
217
Consequently,
L can be chosen so that(P0),..., (P5)
all hold. With such a choice of L wehave,
for all(xl, x2 )
in a dense open subset ofD03C0:
Because of
(*)
these conditionsimply
condition(c),
above. But condi- tions(a)
and(b)
also hold for all(xl, x2 )
in a dense open subset ofD03C0.
Therefore,
X’ istransversally biplanar (in
theanalytic sense)
at everypoint
of some dense open subset of V.Q.E.D.
4.
Seminormality
ofhypersurf aces
withnonsingular
normalization Let X be acomplete nonsingular variety
of dimension r. Let 7r: X ~ X’be a finite birational
morphism,
where X’ is ahypersurface
inPr+1k.
ThusX is the normalization of X’. Moreover there is an exact sequence of sheaves on X’ :
Let W be the conductor of
(9x,
in03C0*OX. By
definition W is thelargest
sheaf of ideals in
(9x,
which annihilates(03C0*OX)/OX’.
The double locus of X’ is the closed subscheme D c X’ whose structure sheaf is(9x,IW.
Thenx E D if and
only
if(9x,@x
is not normal. Since X’ hasnonsingular normalization, x E
D if andonly
if x is asingular point
of X’. D is aCohen-Macaulay
scheme of pure dimension r - 1[13,
Theorem3.1].
Since qr is an affine
morphism,
theconductor L ~ OX’
lifts to a sheaf ofideals in
(9x [6,
page163].
Let A be the closed subscheme of X whose structure sheaf is(9xl W.
Then à =03C0-1(D).
à is called the inverseimage of
the double locusof
X’by
03C0, or, the double locusof
’1T.PROPOSITION 4.1: With the notation as above we have:
(a)
X’ is a seminormalvariety if
andonly if à
is a reduced subschemeof
X.
(b) If 0 is seminormal,
then D is seminormal.PROOF:
Being
reduced andbeing
seminormal both are localproperties.
Let
V = Spec(A) be
an affine open subset ofX’,
and let U =03C0-1(V) = Spec( 1),
where A is theintegral
closure of A. Let C be theconductor
ofA. Then
(a)
amounts tosaying
that A is seminormal if andonly
ifA/C
isa reduced
ring.
Since A isS2,
the assertion of(a)
followsby [14,
Lemma1.3]
and[5, Corollary 2.7].
To prove
(b),
we show that D is coveredby
seminormal affine open sets. The scheme structures of D ~ V and 0 n U aregiven by
C taken asan ideal of A and A
respectively. By (a),
A is seminormal. ThusAIC
isreduced.
AIC
is afinitely generated k-algebra,
henceAIC
is aMori ring
[17,
Vol.I, Page 267,
Theorem9]. A is
a finiteA-module,
thusAIC
is anoverring of A/C
which is afinite A/C-module. Therefore by [5, Proposi-
tion
2.5] AIC
is seminormal inA/C. By assumption AIC
is seminormal.Thus if we show
that
theintegral
closure ofAIC
is a subset of theintegral
closure ofAIC, AIC
would beseminormal,
and hence theseminormality
of D willfollow.
Let C= Q1
r1 ... ~Qn
be the minimalprime decomposition
of C inA.
Then C= ( Qj ~ 4)
n ... ~( Qn ~ A) = Pl
~ ...
r1 Pm
is theminimal prime decomposition
of C in A for somem n. The map
A/P ~ A/Q
isinjective whenever Q
lies over P. Thuswe have the
following
commutativediagram:
where
k(Pl)
is the residue fieldAp/PIAp,
and so on. If x E03A0ml k(Pl)
and x is
integral
over03A0ml=1(A/Pl),
then x E03A0nl=1k(Ql)
and itis integral
over
11" i(AIQ,).
But since(A/C)_
=03A0ml=1(A/Pl)-
and(A/C)- - 03A0nl=1(A/Ql)-,
we have(A/C)-
c(A/C)-,
asrequired.
COROLLARY 4.2: In
Proposition
4.1 let r = 2.if à
isseminormal,
thenD c
p3 is a
curve withonly
nodes andtriple points
with threeindependent
tangent lines.
PROOF: This follows
by [3, Corollary 4].
Observe also that theonly singularities
of à can benodes,
because X is anonsingular surface,
andthus at every
point
there is aunique tangent plane,
and thetangent
linesto the curve à lie on this
plane,
and arelinearly independent.
EXAMPLE 4.3: It is well known that if in
Corollary
4.2 03C0 is ageneric projection,
then D is seminormal and itssingularities
areonly ordinary triple points.
Howeverby
thefollowing simple example
D may have nodes when qr isjust
a finite birationalmorphism.
Let X’ be the surface
given by
thefollowing polynomial
over the fieldof
complex numbers,
219
Figure 4.3.1.
F is irreducible
in C[x, y, z].
At P =(0, 1, 0),
X’ has twoanalytic branches,
and itlocally
looks like the union of thehyperboloid 1 - y2 +
z2 - x 2 =
0 and theplane 1 - y = 0 (Fig. 4.3.1).
A direct calculation ofaF/ax, aF/a y, aflaz
shows thatSing(X’)
is the union of the two lines{x
± z =0, 1 - y = 0}.
Since the normalization of X’ isnonsingular,
D isthe union of these two
lines,
and hence D has a node at P. Let F =FI F2
be the factorization of F
in C [[ x, 1 - y, z ]],
thenX’,P~C[[x,1 -
y,
z]]/(F1F2).
The conductor ofX’,p
as an idéal ofmx’,p
is( Fl , F2)/(FI F2).
Thus as an idealof C[[x, 1 - y, z]]/(F1)
XCUx,
1 -y,
z]]j(F2),
the normalization ofX’,p,
the conductor is( Fl, F2)/(F1)
X( Fl , F2)/(F2). By isomorphism
of power seriesrings,
wehave C[[x,
1 -y,
z]]/(F1, F2) ~ C[[x, z]Jj (z 2 - x2).
This means that if we let03C0-1(P) = {Q1, Q2},
then 0 has a node atQI
and another node atQ2.
Inparticular
à is seminormal.
References
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(Oblatum 21-VII-1982)
Joel Roberts
School of Mathematics
University of Minnesota
Minneapolis, MN 55455
USA
Rahim Zaare-Nahandi
Department of Mathematics
Faculty of Sciences
University of Tehran
Tehran 14 Iran