Notes in Math. 340, Springer-Verlag, 1973. [4] Ein L., Varieties with small dual varieties, I, Invent. Math. 86 (1985), 63-74. [5] Fujita T., Impossibility criterion of ...
Some P r o p e r t i e s "of I ~ a l V a r i e t i e s and Their A p p l i c a t i o n s i n P r o j e c t i v e
Geometry
F. L. 2ak Central Economics Mathematical
Institute
Acad. Sci. USSR, Krasikov st. 32 Moscow 117418, U.S.S.R. In this lecture I consider three different problems in complex projective geometry whose statement and/or solution involves the notion of dual variety. Let xnc~ N be an n-dimensional ~X={(x,~)~Sm X,
xx~Nll~x~TX, } , _x
complex projective variety,
and let
where Sm X is the set of nonsingular points of
TX,x
is the tangent space to X at x, @N is the dual projective space, ± N vN is the hyperplane in P corresponding to ~ , and the bar denotes the
closure in
X×@N.
projections.
Let
P:~X
)X
and
~:~X
)@N ~:~X
>~N be the natural
The variety ~=~(PX)C~ N is called the dual variety of X in
The famous blduality
(reflexivity)
~N.
theoFem [which generally fails for V
varieties defined over fields of positive characteristic] Under different guises
states that ~=X.
(e.g. Legendre transforms] dual varieties have been
considered in various bFanches of mathematics for over a hundred years, and the biduality theorem essentially rephrases the well known duality between the Lagrange and Hamilton-Jacobi
approaches in the classical mechanics.
From now on let X be a nonsingular variety. By definition, points of the dual variety ~
the
are in a natural one-to-one COFrespondence
with the tangent hypeFplanes to (or the singular hyperplane sections of) X, and various kinds of geometrically meaningful unusual behavior of hyperplane sections manifest themselves more explicitly in terms of dual varieties.
Thus it makes sense to consider some natural invariants and
properties of dual varieties and see how they reflect in geometric properties of original varieties.
The simplest invariant of ~ is its
dimension ~. It is known that for n=l OF 2 we have ~=N-I and for nZ3 we v
naN-n+l with equality holding if X is a scroll over a curve with fibeFs ~n-Ic~N [18], [8], [4], [9]. Furthermore, ~Zn [18] and if n=n~2N/3, have
V
then there are the following possibilities: in p3; X=plx~n-Ic~2n-1
X is a curve in ~2 OF a surface
(SegFe embedding); X=G(4,1)6c~ 9 (Grassmann variety
of lines in p4); X=SI0c~I5
(spinor variety of four-dimensional
subspaces on a nonsingular eight-dimensional that if
~=na2N/3, then
X n is a hypersurface).
quadrlc)
linear
([4]; it is plausible
Ein has also established some
other geometric properties of nonsingular varieties with small dual
274
varieties.
Anyhow, for most varieties ~ is a hypersurface;
is not a hypersurface and
X'=Xc~, ~e~N is
moreover,
if
a generic hyperplane section of
X, then ~' is the projection of ~ from the point =, and so codimXY=codimXZl Thus, taking linear sections, we can always reduce our problem to the case ~=N-I via the techniques of projective extensions After the dimension,
(see below).
the second natural invariant of dual variety is
its degree. We define the code~ree codegX by the equality codegX=deg~. is a hypersurface, classical
If
then codegX is Just the class of X; this is a
lnvariant playing a very important role in enumerative geometry.
If ~ is not a hypersurface and X' is a generic hyperplane section of X, then it is clear that
codegX'=codegX.
With regard to codegree,
the most simple nonsingular projective
varieties are those whose codegree is small. The problem of classifying varieties of small codegree is parallel to that of classifying of varieties of small degree. Much is known about this last problem. The case of varieties of degree two is classical.
Over thirty years ago A. Well gave a
complete description of varieties of degree three [xxx]. Fifteen years ago Swinnerton-Dyer
succeeded in classifying all varieties of degree four [15].
After that, due particularly
to Ionescu [6], there was considerable
progress in classification of varieties of small degree,
and now we have a
complete list of nonsingular varieties whose degree does not exceed eight. More generally,
Hartshorne,
Barth, Van de Ven and Ran (see [10]) proved
that if degree is sufficiently small with respect to dimension,
then our
variety is a complete intersection. It seems worthwhile to consider similar problems for codegree, but the situation here is quite different.
For example,
in the case of
varieties of small degree one can proceed by induction using the fact that a general hyperplane section has the same degree, whereas there is no such inductive procedure for oodegree.
Furthermore,
while there always exist
varieties of a given degree and arbitrary dimension
(e.g. hypersurfaces),
one can go as far as to ask whether for a given natural number d>2 there exists a natural number n(d) such that each nonslngular variety X with codegX=d one has dimXsn(d).
Of course,
this is not so for d=2 since the
only varieties of class two are quadrics. However already for d=3 we have n(3)=16. Moreover,
one can prove the following analogue of Weil's result.
Theorem. There exist exactly ten non-degenerate
(i.e. not lying in a
hyperplane) nonsingular complex projective varieties of codegree three, namely the self-dual Segre threefold PIxp2cp5,
its hyperplane section FICp4
obtained by blowing up a point in p2 by means of the
linear system of
conics passing through this point, the four Severl varieties,
viz the
275
Veronese surface variety
v2(P2)cp5 , the Segre variety
G ( 5 , 1 } 8 c ~ 14 o f l i n e s
i n E5 a n d t h e v a r i e t y
the orbit
of highest
algebraic
g r o u p E6, a n d t h e f o u r v a r i e t i e s
Severi
varieties
weight vector
from generic
Veronese surfaces
over composition
space corresponding
3x3-matrices variety
algebra
X=v2(P ~)_ a n d ~ i s d e f i n e d
~,
in accordance
with Gelfand's
equation
of X as a suitable
everything
More p r e c i s e l y ,
consider
the
space of Hermitian and let
approach,
X be the projective
det=O.
one c a n i n t e r p r e t
theorem.
apply Well's
Thus d e g the
of Z has multiplicity
If ~ is not a
classification
of X with a generic
t o t h e c a s e when ~ i s a ( s i n g u l a r }
Then e a c h p o i n t
as
determinant.
t h e n we c a n e i t h e r the intersection
spaces.
whose r a n k d o e s n o t e x c e e d o n e . Then
A few w o r d s a b o u t t h e p r o o f o f t h i s
or consider
the
can be interpreted
in the dual space by the equation
X=3 a n d ,
hypersurface,
ambient projective
to
of the
by projecting
varieties
algebras.
to the matrices
representation
obtained
of their
Severi
the Grassmann
E I 6 c p 26 c o r r e s p o n d i n g
the standard
to the vector
over a composition
corresponding
for
points
It should be mentioned that
projective
?2x~2cpS,
hyperplane
c u b i c i n ~N.
two, and s i n c e
theorem to and reduce L e t Z = S i n g X.
deg X=3 we c o n c l u d e
that SZC_~, where SZ is the variety of secant of the (possibly singular) projective variety Z. Let x be a generic point of X, let ~x=p-l(x), and let Zx={~e~xlX is not a non-degenerate quadratic singularity of easy to see that Z ~ x
I~-X}.
~
is either a hyperplane or a quadrie in ~ x
first case it is possible to show that Z is a linear subspace in generally, one can prove that if Z=Sing~ is linear, then X=F scroll and Z=s ±, where scE e is the minimal section. in our situation this means that X=E I.
It is . In the .
Quite
is a rational
e Since codeg~e=deg~ e,
If Z x is a quadrlc,
then SZ=~ and
one can show that Z is nonslngular and either dlmZ=n or dimZ=n-l,
in the
first case X is a Severi variety, and in the second case X is a nonsingular projection of a Severi variety from a point. The next question to ask about dual varieties concerns the nature of their singularities. The simplest case is when ~ is smooth. But, as we have already pointed out, in this case ~=n and all such varieties were classified by Ein (provided that n~(2/3)N). The next natural condition for is normality, and it turns out that this condition yields interesting implications for the geometry of X. To illustrate this point, we first consider the problem of projective extensions of smooth projective varieties. A variety xncp N is called pro]ectlvelv extendable if there exist a variety (x')n+Ic~ N+I and a point ~e~ N+I such that I~-X'=X and cone (i.e. ~' is non-degenerate).
X" is not a
In this case X' is called a projective
276
extension of X. It should be noted that in general X may be extendable in many different ways for which the lengths of maximal chains of successive projective extensions may also be different. We illustrate this point by a simple example. Let
X=v4(Pl)c~4 be
the rational normal curve of degree
four. On the one hand, X is a hyperplane section of the Veronese surface vZ(~2) which is itself non-extensible (cf. below). On the other hand, X is a hyperplane section of the two distinct surfaces ~2c~5 and ~IxQl=Segre ~ 2(~I × ~ I) . Both F_ and ? 1xQ 1 are hyperplane sections of a 2 Zl scroll X"cP-'with fiber ~ over ? which in its turn is a hyperplane section of the Segre variety pIxp3cpT; generic nonsingular sections of ~Ixp3 by pSc~Tare isomorphic to ~ixQl while special sections are isomorphic to F 2. Another useful notion is that of smooth extendabilit Y. A nonsingular variety xncp N is called smoothly extendable if there exist a nonsingular
(x')n+Ic?N+I
variety
and a point ~c~ N+I such that ia.X'=X. Clearly, smooth
extensibility implies extendability, but the converse is generally (and even usually) false. The criterion of non-extendability given below together with Sommese's well known results on non-exlstence of smooth extensions allow to construct many examples illustrating this point (e.g. product varieties). However here we prefer to give a more exotic example due to Fano, Iskovskih and L vovskll. Let_ X'=v~(S._ z z J(v~(~2)))' where vm Sz(Vs(~Z))c~IU is the cone with vertex v3(P2)¢?9.
Then KX,=OX,(-I) and
with a unique singular point X'. Then
HI(X,Ox)=O,
is the Veronese map of order m and z~over__ the Del Pezzo surface
X'c? 38 is a Fano threefold of degree 72
Let X2
