Michał Germaniuk
Projective subvarieties of the Pl ü cker Variety and Rational Morphisms
Introduction
Let f : X Pn be a morphism from a variety X Pn to a projective space Pn .over field K. If for some set of lines of Pn almost every line (i. e.outside the set of codim nonzero) is of the form [x , f(x)]* ( x X ) we refer to the set realizes the mapping f.
In this paper the relationship between rational morphisms, their realizations by the algebraic set of lines and corresponding to them subvarieties of Plücker quadric is presented.
This connection is used to prove some theorems which provide more informations about rational morphisms and to get interesting conclusions. Especially using our methods we obtain theorems about birational morphisms of projective line. Geometric constructions of rational mappings are examined in [4] , [6] , [7] , [8] , [9].
In the above references some examples of our general point of view can be also found.
We assume throughout this paper that the field K is algebraically closed.
1. Preliminaries.
1.1. Projective sets and rational mappings
.The projective set in the n - dimensional projective space Pn is the set of zeros of a finite numbers of forms Fi( x0, x1, ..., xn ) = 0, i = 1, 2, ..., k .
Taking in Pn projective sets as closed sets we obtain a topology called the Zariski topology in Pn .
Theorem 1.1.1
The set X Pnx Pm is closed if and only if it is the set of zeros of the system of equations:
Gi( x0, x1, ..., xn; y0, y1, ..., ym ) = 0, i = 1, 2, ..., k where: Gi is a biform of bidegree (n+1,m+1) .
Proof: Monograph [ 1 ] s.69
The rational morphism f : XPn from a projective set X Pn is presented in the form
f(x) = ( F0(x):F1(x): ...:Fn(x)) ( 1.1.2 )
* In the paper by the symbol [ a , b ] denotes a line passing though te points a , b.
The rational mapping f is regular for x X if there exists a representation ( 1.1.2 ) in which for some 0 i n Fi(x) 0. The rational mapping is regular on X if it is regular for all x X .
Theorem 1.1.3
The image of the projective set in a regular mapping is a closed set.
Proof: Monograph [ 1 ] s.70
1.2 Plücker cordinates of lines
.Let Ln denote the set of all lines in Pn . By Qn we denote the mapping from Ln ( i.e. the Plücker mapping of lines ) defined in the following way : for p Ln .
Qn(p) = ( p01: p02: ...: pkl: ...: p(n-1)n)
for each 0 k < l n , where: pkl = xkyl - xlyk for any x,y p; x y ; x = ( x0: x1: ...: xn ) ; y = ( y0: y : ..: yn)
The mapping Qn is well defined (i.e.does not depend on the choice of the points on the line ), single-valued and "onto" the projective set
PQn Pm , where m = n + 1 2 1
determined by the equations:
pijpkl - pikpjl + pilpjk = 0 , 0 i < j < k < l n
The homogeneous coordinates pij ; 0 i < j n in Pm are ordered lexicographically.
The homogeneous coordinates of the image of the line in this mapping will be called the Plücker coordinates of the line and the projective set PQn - the Plücker variety of lines.
The subset Ln is called an algebraic set of lines if and only if Qn () is a irreducible projective set of PQn.
1.3 Properties of the Plücker mapping of lines.
a). The image in the mapping Qn of the pencil of lines with the vertex x = ( x0 : x1: ...: xn) (the set of lines passing through the point x), is the n-1 dimensional linear set in Pm
determined by the equations:
xkplm - xlpkm + xmpkl = 0, 0 k < l < m n ( 1.3.1 ) b). Let us fix the parameters p01, p02,...,p0n, p12 , ..., p(n-1)n in the system equations ( 1.3.1 ). If those parametrs are homogeneous coordinates of the point P PQn then this system with respect to the variables x0, x1, ..., xn determines in Pn a line p such that Qn(p) = P .
c). Let us fix the parameters p01, p02, ..., p0n, p12, ..., p(n-1)n of equations (1.3.1 ). If those parameters are the homogeneous coordinates of a point P PQn then this system with respect to the variables x0, x1, ..., xn has only the zero solution.
2. Projective subvarieties of the Pl ü cker Variety PQ
nof Rational Morphisms
2.1 Construction of an assignment
.Let X Pn be the projective set determined by the system of equations:
Hi(x0, x1, ..., xn) = 0, i = 1,2, ..., k
and let f :X Pn be the rational mapping which can by presented in the form ( 1.1.2 ) i.e.
f(x) = ( F0(x) : F1(x) : ... : Fn(x) ) where: F0,F1, ..., Fn are formes of the same degree.
Let us consider the set of zeros of (n , m) - biforms:
Hi(x0,x1, ..., xn ) = 0 i = 1, 2, ..., k
xkplm - xlpkm + xmpkl = 0 0 k < l < m n ( 2.1.1 ) Fkplm - Flpkm + Fmpkl = 0 0 k < l < m n
Because the left hand side of these equations are (n , m) - biforms, from Theorem 1.1.1 it follows that they describe a closed projective set in Pnx Pm, where m = n + 1
2 1
. We will denote it by EXf .
Because the projection 2 : Pnx Pm Pm, 2(x ; y) = y for x Pn i y Pm is a regular morphism by Theorem 1.1.3 it follows that the image 2(EXf) = Xf ' Pm is a closed projective set. From the system of equations ( 2.1.1 ) and from the properties 1.3 a).,b).,c). we obtain Xf ' PQn .
The projective set Xf ' depends on the representation of the mapping f. Taking Xf as an intersection of all projective sets Xf ' for different representations of f we obtain the uniquely defined projective set.
2.2 Properties of the projective set Xf .
Let us introduce three sets of lines:
Xfw is the image in the mapping Qn of the set of all lines of the form [ x , f(x) ] where x is a regular point of the rational mapping f and f(x) x ,
Xfs is the image in the mapping Qn of the set of all lines passing through the fixed points, of f ( x = f(x)),
Xfn is the image in the mapping Qn of the set of all lines passing through the singular points of f..
From the system of equation 2.1.1 and from the properties 1.3 a), 1.3 b), 1.3 c) we obtain:
Corollary 2.2.1
Xf = Xfw Xfs Xfn . ( 2.2.2 ) . We note also
Corollary 2.2.3
The sets Xfn , Xfsn = Xfs Xfn are projective sets.
Proof: By notation in the paragraph 2.1 , adding to the system (2.1.1) the equations xiFj(x) - xjFi(x) = 0, 0 i < j n, and going on analogously as in 2.1 we obtain the projective set equal to Xfsn . Similary, if we add to the system (2.1.1) the equations:
Fi(x) = 0 i = 0,1, ...,n , we obtain the projective set equal to Xfn .
By the symbols Xfr and Xfdp we will denote the closure of the set Xfw and Xfr \ Xfw in the Zariski topology ( Xfr = Xfw ; Xfdp = XfrXfw ).
With this notations we get Corollary 2.2.4.
The algebraic set of lines Qn-1(Xf r) realizes the rational mapping f : X Pn.
Proof: This corollary follows from the decomposition (2.2.2) and from the fact that the projective set Xfdp in Xfr has a nonzero codimension.
The projective set Xfr can be obtain directly as a closure of the set Xfw in the Zariski topology without the knowledge of the decomposition (2.2.2).
Moreover Xfw = H (X), where H : X PQn is the rational mapping defined as follows:
if the polynomials F0,F1, ..., Fm determine a mapping f in the form (1.1.2), then for. x
= ( x0 : x1: ..: xn) X, pij = xiFj(x) - xjFi(x), 0 i < j n .
The projective sets Xf and Xfr of the Plücker variety assigned to the rational mapping f for which the algebraic set of lines Qn-1(Xf r) realizes this mapping provide the additional possibilities in the geometric investigations of rational mappings. If we had used only the projective set Xfr without the knowledge of Xf and its decomposition (2.2.2) we would not be able to obtain the following interesting result.
Corollary 2.2.5
For the line p Ln if Qn(p) Xfr and Qn(p) Xf w then Qn(p) Xf sn. Proof: From the decomposition (2.2.2) we have Xf w Xf.
Hence Xfr = Xfw Xf = Xf or Qn(p) Xf. Using again the decomposition (2.2.2) and the fact Qn(p) Xfw we have : Qn(p) Xfs Xfn = Xf sn .
From Corollary 2.2.5. it follows that every line belonging to the algebraic set of lines realizing the rational mapping f of the projective set X is determined by a point x X and by its image different from x or passing through the fixed or singular point of f.
The essential application of Corollary 2.2.5 will be presented in the next paragraph and in other papers from this series.
To complete the properties of Xf we can easy see that:
Corollary 2.2.6
1 dim Xf r dim X
2 If the projective set X is irreducible then the projective set Xf r is irreducible ,too.
3. Rational mappings of the line in P
2 .Example 3.1
Let X be a line in P2 determined by the equation x2 = 0 and let f : XP2 be the rational mapping defined in the following way :
f( x0, x1, x2 ) = ( a00x0 + a01x1 , 0 , a10x0 + a11x1 ) where: a00a11 - a10a01 0 .
Taking into consideration x2 = 0 we see that the system of equations ( 2.1.1 ) can be written in the form:
x0p12 - x1p02 = 0
( a00x0 + a01x1 ) p12 + ( a10x0 + a11x1 ) p01 = 0 After some transformation we obtain :
1 If a10 0 then Xf is determined by the equation:
a10p01p02 + a11p01p12 + a00p02p12 + a01p212 = 0
2 If a10 = 0 then Xf is the union of Xfr and Xfs determined by the equations : a01p12 + a11p01 + a00p02 = 0 and p12 = 0 respectively.
The algebraic set of lines Q2-1(Xfr) has the form, respectively:
1 The set of lines tangent to the conic defined by the equation:
a102x02 + a112x12 +a002x22 -2a11a10x0x1 - 2a00a10x0x2 + 2(2a10a01 - a11a01 )x1x2= 0 2 The pencil of lines with vertex x = ( a01:-a00 :a11 ) .
Lemma 3.2
Let f :X P2 be a rational mapping, where X is a line and f(X) does not include in X . Then :
a) Xfr is irreducible curve for which the Q2(X) has the order n-1, where n = deg Xfr . b) The algebraic set of lines Q2-1(Xf r) consists of lines tangent to a certain curve Kf with degree m and class n , where m is a class of the curve Xfr and n = deg Xfr . The line X is tangent to the curve Kf in n-1 points.
Proof: The property that Xfr is irreducible curve follows from Corollary 2.2.6 and from the fact that f(X) does not include in X. It remains to show that the point Q2(X) is of order n-1. Then the property b) follows from the well known duality.
Let us suppose that the point Q2(X) is of order n-k , where k 2. In this case almost every line passing through this point intersect the curve Xfr at least in two points different from Q2(X). It means that for almost every point x X there exist two lines belonging to the algebraic set of lines Q2-1(Xf r) and passing through this point different from the line X.
It contradicts corollary 2.2.5 from which follows that for regular and nonfixed point x may exist at most one such a line. Also, the order of the point Q2(X) can not be greater then n-1. Otherwise n = 1 , but in this case the pencil of lines with vertex on the line X can not realize the rational mapping.
assumptions the projective set Xf is determined by one equation
F(p01, p02,p12 ) = 0 . If x = ( x0 : x1: x2 ) is a fixed point ,then the order of this point is defined as the maximum of integers k such that the polynomial F is divisible by the polynomial
wk( p01, p02, p12 ) = ( x0p12 - x1p02 + x2p01 )k .
It is clear, that the order of the fixed point is greater or equal one. In the next part of the paper the fixed point of order k > 1 will be treated as k fixed points of order 1.
Lemma 3.3
Let the assumptions of Lemma 3.2 be fulfiled and let n = deg Fj, where Fj, j = 0,1,2 be formes determining the rational mapping f , such that for each point
x X, (F0(x) : F1(x) : F2(x)) 0*. Then:
deg Xf r = n + 1 - k where k denotes the number of fixed points of morphism f
Proof: Without loss of generality we may suppose that the line X is determined by the equation x2 = 0. Because x2 = 0, after some transformations of the system (2.1.1) we obtain the equation of the set Xf in the form:
F(p01, p02, p12) = F0(p02, p12)p12 - F1(p02, p12)p02+F2(p02, p12)p01= 0
If a point x = ( x0 : x1: x2 ) is fixed for the mapping f, then according to the decomposition (2.2.2) the line defined by the equation x0p12 - x1p02 = 0 as the image of the pencil of lines with vertex x in the mapping Q2 is included in Xf. Hence:
F(p01, p02, p12) = H(p01, p02, p12) (x01p12 -x11p02 )...(x0kp12 -x1kp02) where: xi = (x0i : x1i : 0) i = 1,2,...,k are fixed points.
Because for each x X , (F0(x) : F1(x) : F2(x)) 0 implies Xf n= 0 , the set Xfr is defined by the equation H = 0 . From this we have:
deg F = n + 1 and deg H = deg F - k or deg Xfr = deg H = n + 1 - k Lemma 3.4
Let the assumptions of Lemma 3.3 be fulfiled and let f be a birational mapping (i.e. for which exists the rational mapping g : f(X)X such that fog = idf(x) and gof = idx ). Then
deg Fi = deg f(X).
Proof: If X1 is any line and h : X1 X is a projective mapping of lines, then the mapping foh :X1 f(X) will be determined by polynomials of the same degree as the polynomials in the mapping f . So, we can limit our consideration to the case when the birational mapping f doesn't have fixed points and the line X intersects the image f(X) in n different points , where n = deg f(X) and every point of intersection has only one inverse image. By these assumptions ( they yield Xfs= Xfn = 0 ) and Corollary 2.2.5 we obtain that the line passing through the point Q2(X) ( it corresponds to the image of the pencil of lines with vertex x X in the mapping Q2) intersect the curve Xf r only in the point Q2(X) if and only if [ x, f(x) ] = X . There are n = deg f(X) points among x X such that [ x, f(x)] = X. Because the point Q2(X) for the curve Xfr is of order m - 1 where m = deg Xfr (see Lemma 3.2) then the line which intersects the curve Xfr only in the point Q2(X) is tangent to this curve. There are n such points and their number is equal to the order of the point Q2(X), so we have n = m - 1. Hence m = deg Xf r = n + 1 = deg f(X) + 1 . From
*for the rational mapping of a line always exists a representation with this property
Lemma 3.3 we have also deg Xf r = deg Fi + 1 ( the number of fixed points is zero ), so deg Fi = deg f(X).
Lemma 3.5
Let the assumptions of Lemma 3.3 be fulfiled and let f be a birational r mapping. The curve Xfr is a line if and only if the points of the set X f(X) are fixed by f and the number of such points is equal to deg f(X).
Proof: If Xfr is a line then the algebraic set of lines Q2-1(Xf r) realizing the mapping f is a pencil with vertex y X . From corollary 2.2.5 follows that each line p from this pencil intersects the curve f(X) at most in two points y and f(x) where x = p X. It means, that the point y is of order n - 1, where n = deg f(X) , so the mapping f is a projection from such a point for which the points of the set X f(X) are fixed. Conversely, if the points of the set X f(X) are fixed in the mapping f , then from Lemma .3 we have deg Xfr = n + 1 - k , where n = deg Fi and k - the number of the fixed points of the set X f(X) . Using Lemma 3.4 and the fact k = degf(X) we have
n = deg Fi = deg f(X) = k or deg Xfr = k + 1 - k = 1 .
Bibliography:
[1] I.R. Szafarewicz: Osnowy algebraicziskoj geometrji. Izdatielstwo Nauka.Moskwa 1972 [2] A. Biernawski: Przestrzeń prostych przestrzeni rzutowej Pn . Zeszyty naukowe nr 139.
Warszawa 1966
[3] E. Pascal: Repetytoryum matematyki wyższej t.II Geometria. Wyd.Red. Wiadomości Matematyczne. Warszawa 1901
[4] L. Cygan: Odwzorowania powierzchni 2 - go stopnia realizowane za pomocą kongruencji liniowych. Warszawa 1971
[5] W. Blaschke: Projektiv Geometrie. Stuttgart 1934
[6] W. Stankiewicz : Rzut przestrzeni z krzywej zdegenerowanej przestrzennej stopnia czwartego. Zeszyty Naukowe. Geometria. Warszawa - Poznań 1967.
[7] H. Adasiewicz, E. Ko¿niewski : Über eina Konstruktion der Kurve dritten Grades mit dem Doppelpunkt, Zeszyty Naukowe. Geometria 1978.
[8] R. Matla : O pewnych osobliwościach kompleksu biegunowego. Zeszyty Naukowe Geometria. Poznań 1977.
[9] E. Koźniewski : Rzut dwubiegunowy przestrzeni trójwymiarowej. Zeszyty Naukowe Geometria. Poznañ 1980.
WARMIÑSKO MAZURSKI UNIVERSITY DEPARTMENT OF MATHEMATICS 10 - 957 OLSZTYN, POLAND
