PREGUNTAS 1 ¿Profesión?
8. ANALISIS DE ENTREVISTAS
If M is non-singular then it has no singular places
and so oj is of the first kind iff it is regular on M.
In this case, co is then regular along any finite place
v of M. Now suppose that X is singular. If the dif
ferential co is of the first kind and v is a finite place
of X then either v is singular, in which case u is regular
along v; or v is a finite place of the smooth locus of X.
The differential w is regular on this locus, by assump
tion, and so again, ui is regular along v. This proves
part of the next result. The remainder follows immedi
ately from Theorem 2, Corollary 3.
Theorem 3 . The differential w is of the first kind
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Corollary. Suppose X is complete. Then the differen
tial to is of the first kind iff it is everywhere
regular.
Even though the variety X may have no differentials
that are everywhere regular, on affine varieties there
are many differentials of the first kind. For example,
if the functions f and g are regular on X then f dg is of
the first kind. More generally, any differential <o
Kahler on X is of the first kind. However, the follow
ing example shows that not every differential of the
first kind is Kahler on X.
Suppose that X be the plane curve defined by the
2 3
equation y = x . It has a birational parametrization
2 3
x = t and y = t . The differential dt is of the first
kind but is not Kahler on X.
Theorem 4 . Suppose X' -*■ X is a resolution of X.
Then the differential co is of the first kind iff it is
regular on X '.
Proof. If co is of the first kind it is regular
along every finite place of X. As every place appearing
in X' is finite, <o is regular on X'.
Conversely, suppose co is regular on X'. As X' + X
is proper, any finite place v of X is a finite place of
X'. As X' is non-singular, co is regular along v.
Corollary. Suppose X' -*• X is a resolution whose excep
tional places are Vj.... v^. Then a differential <o regular
on X is of the first kind just in case it is regular
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If the variety X admits a resolution then there is
a finite collection v^,...,v^ of places such that any
differential co regular on X which is also regular along
v^,...,v^ is regular on any other singular place v of X.
To find such a finite collection of places is a weakened
form of the problem of resolution of singularities. At
the end of this section we will associate to X a collec
tion of places, which is finite if the singularity
admits a resolution.
2. Small resolutions
The exceptional locus E of a resolution X' -*■X
need not be a divisor. An extreme case is where every
component of E has codimension at least two. In this
case we say that the exceptional locus of the resolution
is small. We will given an example of a small
resolution.
Let (r,s,t,u) be co-ordinates on /A^ and let X be
defined by the equation rt = su. It is easily seen
that the origin is the only singular point of X. Now
suppose that (v:w) are homogeneous co-ordinates for P'*'
and that X' is the subvariety of x P 1 defined by
the equations vt = wu and vs = wr in addition to
rt = su. Let it be the natural projection from X' to X.
First we show that X' is irreducible and non-
4 1
singular. On the portion of /A * p upon which v is
non-zero, the equations of X' are
w w
t = — • u s - — • r rt = su
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and as the third equation is a consequence of the first
two the result follows. (The case of w non-zero is
exactly similar.)
If the point P = (r,s,t,u) is a non-singular
point of X then one of the co-ordinates, say r, is
non-zero. In this case the point P' = (r,s,t,u) * (r:s)
is the unique point of it ^ (P) . This shows that X ' -*■ X
is a resolution.
The exceptional locus of tt is (0,0,0,0) x P"*".
Further results concerning small resolutions may be
found in [28].
3. Order of differentials
We have just seen that a differential co regular on
X must satisfy additional conditions in order to be of
the first kind, and that if X* -*■ X is a resolution then
these conditions are finite in number. More exactly,
it is enough that oj be regular along the exceptional
places of X'.
Contrarywise, a differential u> regular on X imposes
conditions that a birational map X' -*■ X must satisfy if
it is to be a resolution. For example, if uj is not of
the first kind but is regular on X' then X' is not a
resolution of X.
The quantity ord^(to), which we are about to define,
refines the notion of u> being of the first kind. We
will prove in this section some initial results. In §3
we will make a deeper study of the properties of this
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Definition. Suppose that v is a place and that to
is a differential form. The largest value of n for which
t w is regular along v, where t is a local equation for
v, is said to be the order of to along v and is denoted
by v(w).
The differential t ncj is regular along v just in
case it is regular on some model 0 of v. This shows
that v(u>) does not depend upon the choice of t, and
also that v(w) is non-negative iff w is regular along v.
Definition. If to is a differential regular on X
then the smallest value of v(<u) as v runs through the
singular places of X, is called the order of u> with
respect to X and will be denoted by ord^fajj.
If v(w) is without a lower bound we write
ord^(u) = -» while if X has no singular places we
write ord^(w) = + «■>.
If the differential cj is regular on X and X' + X is
a resolution then the behaviour of w on X' is sufficient
to determine whether or not uj is of the first kind. A
similar result holds for the order of w. Clearly, ou is
Proof. If ord^(ii)) is negative then g o is not of the
first kind and so fails to be regular along some excep
tional place v of X'.
Now suppose that ord^(co) is non-negative. Let n
denote the smallest value of v '(g o) for v* an exceptional
place of X'. Clearly, n is non-negative. It is enough
to show that if v is a singular place of X then v(to) n.