Quantitative properties of the non-properness set of a polynomial map
aa r X i v : . [ m a t h . AG ] N ov Degree of K -uniruledness of the non-properness set of apolynomial map Zbigniew Jelonek and Michał Lasoń
Abstract.
Let X ⊂ C n be an affine variety covered by polynomial curves,and let f : X → C m be a generically finite polynomial map. In [ ] the firstauthor showed that the set S f , of points at which f is not proper, is coveredby polynomial curves.We generalize this result to the field of real numbers. We prove thatif X ⊂ R n is a closed semialgebraic set covered by polynomial curves, and f : X → R m is a generically finite polynomial map, then the set S f is alsocovered by polynomial curves. Moreover, if X is covered by curves of degree atmost d , and the map f has degree d , then the set S f is covered by polynomialcurves of degree at most d d . For the field of complex numbers we get abetter bound by d d , which is best possible.Additionally, if X = C n or X = R n , and a generically finite polynomialmap f has degree d , then the set S f is covered by polynomial curves of degreeat most d − . This bound is best possible.
1. Introduction
Let f : X → Y be a generically finite polynomial map between affine varieties. Definition . We say that f is finite (proper) at a point y ∈ Y , if thereexists an open neighborhood U of y such that f | f − ( U ) : f − ( U ) → U is a finitemap. The set of points at which f is not finite (proper) we denote by S f . This set was first introduced by the first author in [ ] (see also [
2, 3, 10 ]).The set S f is a good measure of non-properness of the map f , moreover it hasinteresting applications in pure and applied mathematics [
5, 6, 9, 11 ]. The firstauthor proved the following property of the set S f , when the base field is C . Theorem ]) . Let X be an affine variety over C , and let f : X → C m be a generically finite polynomial map. Then (1) the set S f is a hypersurface in f ( X ) or it is empty, (2) if X is C -uniruled (covered by polynomial curves), then the set S f is also C -uniruled. Mathematics Subject Classification.
Key words and phrases. affine variety, semialgebraic set, K -uniruled set, the set of non-properpoints, parametric curve, degree of a curve.Z. Jelonek is supported by Polish National Science Centre grant no. 2013/09/B/ST1/04162.M. Lasoń is supported by the Polish Ministry of Science and Higher Education Iuventus Plusgrant no. 0382/IP3/2013/72. One of the most important results of this paper is a generalization of Theorem1.2 to the field of real numbers. Suppose X is a closed semialgebraic set over R ,and f : X → R m is a generically finite polynomial map. Then the set S f does nothave to be a hypersurface, thus part (1) of Theorem 1.2 is false over R . We provethat part (2) is true. That is: Let X be a closed semialgebraic set over R , and let f : X → R m be a genericallyfinite polynomial map. If X is R -uniruled, then the set S f is also R -uniruled. (Theorem 4.9)In another bunch of results we give upper bounds on degree of K -uniruledness(that is, degree of polynomial curves covering the set, see Section 2 Preliminariesfor relevant definitions), for K = C or K = R , of the set S f in terms of degree ofthe map f : Let f : C n → C m be a generically finite polynomial map of degree d . Then the set S f has degree of C -uniruledness at most d − . (Theorem 3.2) Let X be an affine variety with degree of C -uniruledness at most d . Let f : X → C m be a generically finite polynomial map of degree d . Then the set S f has degreeof C -uniruledness at most d d . (Theorem 3.5) Let f : R n → R m be a generically finite polynomial map of degree d . Then the set S f has degree of R -uniruledness at most d − . (Theorem 4.4) Let X ⊂ R n be a closed semialgebraic set with degree of R -uniruledness at most d ,and let f : X → R m be a generically finite polynomial map of degree d . Then theset S f has degree of R -uniruledness at most d d . (Theorem 4.9)All above bounds, except the last one, are best possible. Since they are true for thefield C , by the Lefschetz Principle they are also true for an arbitrary algebraicallyclosed field K with characteristic zero.In the last Section, as an application of our methods, we give a real counterpartof Theorem 3.1 from [ ]. Let G be a real, non-trivial, connected, unipotent group, which acts effectively andpolynomially on a closed semialgebraic R -uniruled set X . Then the set F ix ( G ) ,of fixed points of G , is also R -uniruled. In particular it does not contain isolatedpoints. (Theorem 5.1)
2. Preliminaries
Unless stated otherwise K is an arbitrary algebraically closed field (the real fieldcase is explained in Section 4). All affine varieties are considered to be embeddedin an affine space.The study of uniruled varieties in projective geometry, that is varieties pos-sessing a coving by rational curves, has a long history. In affine geometry it ismore natural to consider parametric curves (see a definition below), than rationalones. Therefore in [ ] (see also [ ]) the first author defined K -uniruled varieties asthose which are covered by parametric curves. In [ ] we refined this definition forcountable fields. In this paper we introduce and study a corresponding quantitativeparameter – degree of K -uniruledness. EGREE OF K -UNIRULEDNESS OF S f Definition . An irreducible affine curve Γ ⊂ K m is called a parametriccurve of degree at most d , if there exists a non-constant polynomial map f : K → Γ of degree at most d (by degree of f = ( f , . . . , f m ) we mean max i deg f i ). A curveis called parametric if it is parametric of some degree. We have the following equivalences, see also similar Proposition 2.4 from [ ]. Proposition . Let X ⊂ K m be an irreducible affine variety of dimension n , and let d be a constant. The following conditions are equivalent: (1) for every point x ∈ X there exists a parametric curve l x ⊂ X of degree atmost d passing through x , (2) there exists an open, non-empty subset U ⊂ X , such that for every point x ∈ U there exists a parametric curve l x ⊂ X of degree at most d passingthrough x , (3) there exists an affine variety W of dimension dim X − , and a dominantpolynomial map φ : K × W ∋ ( t, w ) → φ ( t, w ) ∈ X such that deg t φ ≤ d . Proof.
Implication (1) ⇒ (2) is obvious. To prove (2) ⇒ (1) suppose that X = { x ∈ K m : f ( x ) = 0 , . . . , f r ( x ) = 0 } . For a point a = ( a , . . . , a m ) ∈ K m and b = ( b , : · · · : b d,m ) ∈ P M , where M = dm − , let ϕ a,b : K ∋ t → ( a + b , t + · · · + b d ,d t d , . . . , a m + b m, t + · · · + b dm,d t d ) ∈ K m be a parametric curve. Consider a variety and a projection K m × P M ⊃ V = { ( a, b ) ∈ K m × P M : ∀ t,i f i ( ϕ a,b ( t )) = 0 } ∋ ( a, b ) → a ∈ K m . From the definition ( a, b ) ∈ V if and only if the parametric curve ϕ a,b is containedin X . Hence the image of the projection is contained in X and contains U , sincethrough every point of U passes a parametric curve of degree at most d . But sincethe projective space P M is complete and V is closed we get that the image is closed,and hence it is the whole set X .Let us prove (2) ⇒ (3) . For some affine chart V j = V ∩ { b j = 1 } the abovemap is dominant. We consider the following dominant map Φ : K × V j ∋ ( t, φ ) → φ ( t ) ∈ X. After replacing V j by one of its irreducible components Y ⊂ K M ( dim( Y ) = s ) themap remains dominant. On an open subset of X fibers of the map Φ ′ = Φ | K × Y areof dimension s + 1 − n , let x be one of such points. From the construction of the set V we know that the fiber F = Φ ′− ( x ) does not contain any line of type K × { y } ,so in particular the image F ′ of the fiber F under projection K × Y → Y has thesame dimension. For general linear subspace L ⊂ K M of dimension M + n − s − the set L ∩ F ′ is -dimensional. Let us fix such L , and let R be any irreduciblecomponent of L ∩ Y intersecting F ′ . Now map Φ ′ | K × R : K × R → X satisfies theassertion, since it has one fiber of dimension (at x ) and the dimension of R is n − . To prove the implication (3) ⇒ (2) it is enough to notice that for every w ∈ W the map φ w : K ∋ t → φ ( t, w ) ∈ X is a parametric curve of degree at most d orit is constant. Image of φ contains an open dense subset, so after excluding pointswith infinite preimages (closed set of codimension at most one) we get an open set U with requested properties. (cid:3) ZBIGNIEW JELONEK AND MICHAŁ LASOŃ
Definition . We say that an affine variety X has degree of K -unirulednessat most d if all its irreducible components satisfy the conditions of Proposition 2.2.An affine variety is called K -uniruled if it has some degree of K -uniruledness. To simplify the notion we say that the empty set has degree of K -unirulednessequal to zero, in particular it is K -uniruled. Example . Let X ⊂ K n be a hypersurface of degree d < n. It is well-knownthat X is covered by affine lines, therefore its degree of K -uniruledness is one. For uncountable fields there is also another characterization of K -uniruled va-rieties, see Theorem 3.1 [ ]. Proposition . Let K be an uncountable field, and let X ⊂ K m be an affinevariety. The following conditions are equivalent: (1) X is K -uniruled, (2) for every point x ∈ X there exists a parametric curve l x ⊂ X passingthrough x , (3) there exists an open, non-empty subset U ⊂ X , such that for every point x ∈ U there exists a parametric curve l x ⊂ X passing through x .
3. The complex field case
In the whole Section we assume that the base field is C . The condition that amap is not finite at a point y is equivalent to the fact that it is locally not properin classical topology sense (there is no neighborhood U of y such that f − ( U ) iscompact). This characterization gives the following: Proposition ]) . Let f : X → Y be a generically finite map betweenaffine varieties. Then y ∈ S f if and only if there exists a sequence ( x n ) , such that | x n | → ∞ and f ( x n ) → y . In particular, for a polynomial map f : C n → C n , y ∈ S f if and only if either dim f − ( y ) > or f − ( y ) = { x , . . . , x r } is a finite set, but P ri =1 µ x i ( f ) < µ ( f ) ,where µ denotes the multiplicity. In other words f is not proper at y if f is not alocal analytic covering over y . Theorem . Let f : C n → C m be a generically finite polynomial map ofdegree d . Then the set S f has degree of C -uniruledness at most d − . Proof.
Let y ∈ S f , by affine transformation we can assume that y = O =(0 , , ..., ∈ C m . From the same reason we can assume that O f − ( S f ) . Due toProposition 3.1 there exists a sequence of points x k → ∞ such that f ( x k ) → O . Letus consider the line L k ( t ) = tO + (1 − t ) x k = (1 − t ) x k , t ∈ C . Put l k ( t ) = f ( L k ( t )) .We can assume that deg l k > , because infinite fibers cover only nowhere densesubset of C n . Each curve l k is given by m polynomials of one variable: l k ( t ) = ( d X i =0 a i ( k ) t i , . . . , d X i =0 a mi ( k ) t i ) . Hence we can associate l k with one point ( a ( k ) , . . . , a d ( k ); a ( k ) , . . . , a d ( k ); . . . ; a m ( k ) , . . . , a md ( k )) ∈ C N . Since for each i when k → ∞ , then a i ( k ) → , we can change the parametrizationof l k by putting t → λ k t , in such a way that k l k k = 1 , for k ≫ (we consider here EGREE OF K -UNIRULEDNESS OF S f l k as an element of C N with Euclidean norm). Now, since unit sphere is compact,it is easy to see that there exists a subsequence ( l k r ) of ( l k ) , which is convergent toa polynomial map l : C → C m , with l (0) = O. Moreover, l is non-constant, because k l k = 1 , and l (0) = O. We can also assume that the limit lim k →∞ λ k = λ exists inthe compactification of the field C . We consider two cases:(1) λ is finite, then L k ( λ k t ) = (1 − λ k t ) x k → ∞ for t = λ − . (2) λ = ∞ , then k L k ( λ k t ) k ≥ ( | λ k t | − k x k k , and hence k L k ( λ k t ) k → ∞ forevery t = 0 .On the other hand f ( L k ( λ k t )) = l k ( λ k t ) → l ( t ) , using once more Proposition 3.1this means that the curve l is contained in the set S f , and as a consequence that S f has degree of C -uniruledness at most d .Now we show that deg l < d. It is enough to prove that deg l < deg l k for some k . Suppose contrary, what is deg l = deg l k for all k . Let l ( t ) = ( l ( t ) , . . . , l m ( t )) and l k ( t ) = ( l k ( t ) , . . . , l km ( t )) . We can assume that a component l ( t ) has a maximaldegree. Denote f ( O ) = a = ( a , . . . , a m ) . All roots of a polynomial l ( t ) − a arecontained in the interior of some disc D = { t ∈ C : | t | < R } . Let ǫ = inf {| l ( t ) − a | ; t ∈ ∂D } . For k ≫ we have | ( l − a ) − ( l k − a ) | D < ǫ. Consequently bythe Rouch´e’s Theorem these polynomials have the same number of zeros (countedwith multiplicities) in D . In particular zeros of l k − a are bounded. All curves L k pass through O , so all l k pass through a = f ( O ) . These means that there is asequence t k such that l k ( t k ) = a . We have just showed that | t k | < R , since t k is aroot of the polynomial l k − a . So we can assume that the sequence t k convergesto some t . When we pass to the limit we get l ( t ) = a , which is a contradiction,since a = f ( O ) S f . (cid:3) In a similar way we can prove the following generalization of Theorem 3.2.
Theorem . Let X = C × W ⊂ C × C n be an affine cylinder and let f : C × W ∋ ( t, w ) → ( f ( t, w ) , . . . , f m ( t, w )) ∈ C m be a generically finite polynomialmap. Assume that for every i we have deg t f i ≤ d . Then the set S f has degree of C -uniruledness at most d . Proof.
Let y ∈ S f , by affine transformation we can assume that y = O =(0 , , ..., ∈ C m . Due to Proposition 3.1 there exists a sequence of points ( a k , w k ) ∈ C × W , such that | ( a k , w k ) | → ∞ and f ( a k , w k ) → y . Let us consider the line L k ( t ) = ((1 − t ) a k , w k ) , t ∈ C . Put l k ( t ) = f ( L k ( t )) . We can assume that deg l k > ,because infinite fibers cover only nowhere dense subset of X . Each curve l k is givenby m polynomials of one variable: l k ( t ) = ( d X i =0 a i ( k ) t i , . . . , d X i =0 a mi ( k ) t i ) . As before we can associate l k with one point ( a ( k ) , . . . , a d ( k ); a ( k ) , . . . , a d ( k ); . . . ; a m ( k ) , . . . , a md ( k )) ∈ C N . Since for each i when k → ∞ , then a i ( k ) → , so we can change the parametrizationof l k by putting t → λ k t , in such a way that k l k k = 1 , for k ≫ (we consider here l k as an element of C N with Euclidean norm). Now, since unit sphere is compact,there exists a subsequence ( l k r ) of ( l k ) , which is convergent to a polynomial map l : C → C m , with l (0) = O . Moreover, l is non-constant, because k l k = 1 and l (0) = ZBIGNIEW JELONEK AND MICHAŁ LASOŃ O. We can also assume that the limit lim k →∞ λ k = λ exists in the compactificationof the field C . We consider two cases:(1) λ is finite, then L k ( λ k t ) = ((1 − λ k t ) a k , w k ) → ∞ for t = λ − . (2) λ = ∞ , then k L k ( λ k t ) k ≥ max(( | λ k t |− | a k | , k w k k ) , and k L k ( λ k t ) k → ∞ for every t = 0 .On the other hand f ( L k ( λ k t )) = l k ( λ k t ) → l ( t ) , using once more Proposition 3.1this means that the curve l is contained in the set S f , and as a consequence that S f has degree of C -uniruledness at most d . (cid:3) Corollary . Let f = ( f , ..., f m ) : C n → C m be a generically finite mapwith d = min j max i deg x j f i . Then the set S f has degree of C -uniruledness at most d . Let us recall (see [ ]) that for a generically finite polynomial map f : X → Y with X being C -uniruled the set S f is also C -uniruled. We have the following“numerical” counterpart of this result: Theorem . Let X be an affine variety with degree of C -uniruledness at most d , and let f : X → C m be a generically finite map of degree d . Then the set S f has degree of C -uniruledness at most d d . Proof.
By Definition 2.3 there exists an affine variety W with dim W =dim X − and a dominant polynomial map φ : C × W → X of degree in thefirst coordinate at most d . Equality dim C × W = dim X implies that φ is generi-cally finite, hence also f ◦ φ : C × W → C m , which is of degree in the first coordinateat most d d . So due to Theorem 3.3 S f ◦ φ has degree of C -uniruledness at most d d . We have an inclusion S f ⊂ S f ◦ φ , and from Theorem 1.2 we know that if notempty, then both sets are of pure dimension dim X − , so components of S f arecomponents of S f ◦ φ . This implies the assertion. (cid:3) Example . Let f : C n ∋ ( x , . . . , x n ) → ( x , x x , . . . , x x n ) ∈ C n . Wehave deg f = 2 and S f = { x ∈ C n : x = 0 } . The set S f has degree of C -uniruledness equal to . It shows that in general Theorems 3.2, 3.3 and Corollary3.4 can not be improved. Example . Let f : C ∋ ( x, y ) → ( x + ( xy ) d , xy ) ∈ C . We have deg f = 2 d and S f = { ( s, t ) ∈ C : s = t d } . The set S f has degree of C -uniruledness equal to d . It shows that in general Corollary 3.4 can not be improved. Example . For n > let X = { x ∈ C n : x x = 1 } , and f : X ∋ ( x , . . . , x n ) → ( x , . . . , x n ) ∈ C n − . The variety X has degree of C -unirulednessequal to , moreover we have deg f = 1 and S f = { x ∈ C n − : x = 0 } . So the set S f has degree of C -uniruledness . It shows that in general Theorems 3.3, 3.5 cannot be improved. Remark . By the Lefschetz Principle all results from this Section remaintrue for an arbitrary algebraically closed field of characteristic zero.
4. The real field case
In the whole Section we assume that the base field is R . Let us recall thatby a real parametric curve of degree at most d in a semialgebraic set X ⊂ R n wemean the image of a non-constant real polynomial map f : R → X of degree at EGREE OF K -UNIRULEDNESS OF S f most d . In general a real parametric curve does not have to be algebraic, but onlya semialgebraic curve. The real counterpart of Proposition 2.2 is the following. Proposition . Let X ⊂ R n be a closed semialgebraic set, and let d be aconstant. The following conditions are equivalent: (1) for every point x ∈ X there exists a parametric curve l x ⊂ X of degree atmost d passing through x , (2) there exists a dense subset U ⊂ X , such that for every point x ∈ U thereis a parametric curve l x ⊂ X of degree at most d passing through x , (3) for every polynomial map f : X → R m , and every sequence x k ∈ X such that f ( x k ) → a ∈ R m there exists a semialgebraic curve W and agenerically finite polynomial map φ : R × W ∋ ( t, w ) → φ ( t, w ) ∈ X such that deg t φ ≤ d , and there exists a sequence y k ∈ R × W such that f ( φ ( y k )) → a . Moreover, if x k → ∞ , then also φ ( y k ) → ∞ . Proof.
First we prove implication (2) ⇒ (1) . Suppose that X = { x ∈ R n : f ( x ) = 0 , . . . , f r ( x ) = 0 g ( x ) ≥ , ..., g s ( x ) ≥ } . For a = ( a , . . . , a n ) ∈ R n and b = ( b , , . . . , b d,n ) ∈ R M , where M = dn , let ϕ a,b ( t ) = ( a + b , t + · · · + b d ,d t d , . . . , a n + b n, t + · · · + b dn,d t d ) be a parametric curve. If there exists a parametric curve of degree at most d passing through a , then after reparametrization we can assume that it is ϕ a,b for P i,j b i,j = 1 . This means that b ∈ S M (0 , , where S M denotes the unit sphere in R M . Consider a semialgebraic set V = { ( a, b ) ∈ R n × S M (0 ,
1) : ∀ t,i f i ( ϕ a,b ( t )) = 0 , ∀ t,j g j ( ϕ a,b ( t )) ≥ } . The definition of the set V says that parametric curves ϕ a,b ( t ) are contained in X . Itis easy to see that V is closed. For any a ∈ X , by the assumption there is a sequenceof points a k → a , such that for every k there is a parametric curve ϕ a k ,b k ∈ V . Wecan assume that k a k k < k a k + 1 for all k . The set V is closed, sequence ( a k , b k ) ⊂ V is bounded, so there is a subsequence ( a k r , b k r ) which converges to ( a, b ) ∈ V . Nowparametric curve ϕ a,b ⊂ X of degree at most d passes through a .We prove (1) ⇒ (3) . Consider a semialgebraic set V as above. We have asurjective map Φ : R × V ∋ ( t, ϕ a,b ) → ϕ a,b ( t ) ∈ X. Let f : X → R m be a polynomial map, and let f ( x k ) → a ∈ R m for a sequence x k ∈ X . Put g = f ◦ Φ . Hence there exists a sequence z k ∈ R × V such that g ( z k ) → a . Due to curve selection lemma there is a semialgebraic curve W ⊂ R × V such that a ∈ g ( W ) . Take W = p ( W ) , where p : R × V → V is a projection.If W is a curve then let W := W , if it is a point we take as W any semialgebraiccurve in V which contains point π ( W ) . Now ( W, Φ | R × W ) is a good pair.Finally to prove (3) ⇒ (2) it is enough to take as f the identity in the thirdcondition. (cid:3) Definition . We say that a closed semialgebraic set X has degree of R -uniruledness at most d if it satisfies the conditions of Proposition 4.1. A closedsemialgebraic set is called R -uniruled if it has some degree of R -uniruledness. Example . Let X = { ( x, y ) ∈ R : x ≥ , y ≥ } . It is easy to check thatthe degree of R -uniruledness of X is two. It has a ruling { ( a, t ) , a ≥ } . ZBIGNIEW JELONEK AND MICHAŁ LASOŃ
Let X ⊂ R n be a closed semialgebraic set, and let f : X → R m be a polynomialmap. As in the complex case we say that it is not proper at a point y ∈ R m ifthere is no neighborhood U of y such that f − ( U ) is compact. The set of all points y ∈ f ( X ) at which the map f is not proper we denote as before by S f . This set isalso closed and semialgebraic [ ]. The results of [ ] can be generalized as follows. Theorem . Let f : R n → R m be a generically finite polynomial map ofdegree d . Then the set S f has degree of R -uniruledness at most d − . Theorem . Let X = R × W ⊂ R × R n be a closed semialgebraic cylinderand let f : R × W ∋ ( t, w ) → ( f ( t, w ) , . . . , f m ( t, w )) ∈ R m be a generically finitepolynomial map. Assume that for every i deg t f i ≤ d . Then the set S f has degreeof R -uniruledness at most d . Corollary . Let f = ( f , ..., f m ) : R n → R m be a generically finite polyno-mial map with d = min j max i deg x j f i . Then the set S f has degree of R -unirulednessat most d . Proofs of these facts are exactly the same as in the complex case.To prove a real analog of Theorem 3.5 we need some ideas from [ ]. Let X be a smooth complex projective surface, and let D = P ni =1 D i be a simple normalcrossing (s.n.c.) divisor on X (we consider only reduced divisors). Let graph( D ) be a graph of D , with vertices D i , and one edge between D i and D j for each pointof intersection of D i and D j . Definition . We say that D a simple normal crossing divisor on a smoothsurface X is a tree if graph( D ) is a tree (it is connected and acyclic). The following fact is obvious from graph theory.
Proposition . Let X be a smooth projective surface and D ⊂ X be a divisorwhich is a tree. If D ′ , D ′′ ⊂ D are connected divisors without common components,then D ′ and D ′′ have at most one common point. Now we are ready to prove a real counterpart of Theorem 3.5. In particular weshow that for a generically finite map f : X → Y od real semialgebraic sets, theset S f is also R -uniruled, provided the set X is so. Theorem . Let X ⊂ R n be a closed semialgebraic set with degree of R -uniruledness at most d , and let f : X → R m be a generically finite polynomialmap of degree d . Then the set S f is also R -uniruled. Moreover, its degree of R -uniruledness is at most d d . Proof.
Let a ∈ S f and let x k ∈ X be a sequence of points such that f ( x k ) → a and x k → ∞ . By Proposition 4.1 there exists a semialgebraic curve W and agenerically finite polynomial map φ : R × W ∋ ( t, w ) → φ ( t, w ) ∈ X such that deg t φ ≤ d , and there exists a sequence ( y k ) ⊂ R × W such that f ( φ ( y k )) → a and y k → ∞ . In particular a ∈ S f ◦ φ . Let Γ be a Zariski closure of W . We canassume that Γ and its complexification Γ c are smooth. Denote Z := R × Γ . Wehave the induced map φ : Z → X . Hence we have also the induced complex map φ c : Z c := C × Γ c → X c , where Z c , X c denote the complexification of Z and X respectively.Let Γ c be a smooth completion of Γ c and let us denote Γ c \ Γ = { a , ..., a l } . Let P × Γ c be a projective closure of Z c . The divisor D = Z c \ Z c = ∞ × Γ c + EGREE OF K -UNIRULEDNESS OF S f P li =1 P × { a i } is a tree. The map φ induces a rational map φ : Z c X c , where X c denotes the projective closure of X c . We can resolve points of indeterminacy ofthis map: π Z cm Z cm − ... Z c X c φ φ ′ π π m − π m ❄❄❄ ✲❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❘ Let us note that H := π − ( Z ) has a structure of a real variety and R × Γ ⊂ H .Let Q := Z m ∩ φ ′− ( X c ) . Then the map φ ′ : Q → X c is proper. Moreover, Q = Z m \ φ ′− ( X c \ X c ) . The divisor D = φ ′− ( X c \ X c ) is connected as acomplement of a semi-affine variety φ ′− ( X c ) (for details see [ ], Lemma 4.5). Notethat the divisor D ′ = π ∗ ( D ) is a tree. Hence the divisor D ⊂ D ′ is also a tree.The map f ′ = f ◦ φ ′ : R × Γ → R m we can consider as the regular map f ′ : Q → C m . This map induces a rational map from H c = Z cn to P m ( C ) . As beforewe can resolve its points of indeterminacy: ψ H ck H ck − ... H c P m ( C ) f ′ Fρ ρ k − ρ k ❄❄❄ ✲❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❘ Note that the divisor D ′ = ψ ∗ ( D ) is a tree. Let ∞ ′ × Γ denote a propertransform of ∞ × Γ . It is an easy observation that F ( ∞ ′ × Γ) ⊂ π ∞ , where π ∞ denotes the hyperplane at infinity of P m ( C ) . Now S f ′ = F ( D \ F − ( π ∞ )) . The curve L = F − ( π ∞ ) is connected (the same argument as above). Now byProposition 4.8 we have that every irreducible curve l ⊂ D ′ (note that necessarily l ∼ = P ( C ) ) which does not belong to L has at most one common point with L . Let R ⊂ S f ′ be an irreducible component. Hence R is a curve. There is a curve l ⊂ D , which has exactly one common point with L , such that R = F ( l \ L ) . If l is givenby blowing up of a real point, then L also has a real common point with l (becausethe conjugate point also is a common point of l and L ). When we restrict to thereal model l r of l we have l r \ L ∼ = R . Hence if we restrict our consideration only tothe real points and to the set Q r := R × W ⊂ Q (we consider here a closure in theeuclidian topology) we see that the set S of non-proper points of the map f ′ | Q r isa union of parametric curves F ( l r \ L ) , l ∈ D , ψ ( l ) ∈ Q r , where the last closureis the closure in a real projective space. Of course a ∈ S ⊂ S f . Similarly the set S f ◦ φ is a union of parametric curves F ( l r \ L ) , l ∈ ψ ∗ ( D ′ ) , π ( ψ ( l )) ⊂ R × W ⊂ Z .Hence we can say that an “irreducible component” of the set of non-proper pointsof f ′ | Q r is also an ‘irreducible’ component of S f ◦ φ . Moreover a ∈ S f ′ | Qr ⊂ S f . Nowwe can finish the proof by Theorem 4.5 and Lemma 4.10. (cid:3)
Lemma . Let ψ : R → R m be a parametric curve. If there exists a para-metric curve φ : R → R m of degree at most d with ψ ( R ) ⊂ φ ( R ) , then ψ ( R ) hasdegree of R -uniruledness at most d . Proof.
Indeed, let φ ( t ) = ( φ ( t ) , ..., φ m ( t )) and consider a field L = R ( φ , ..., φ m ) .By the L¨uroth Theorem there exists a rational function g ( t ) ∈ R ( t ) such that L = R ( g ( t )) . In particular φ i ( t ) = f i ( g ( t )) . In fact we have two induced maps f : P ( R ) → P m ( R ) and g : P ( R ) → P ( R ) . Moreover, f ◦ g = φ. Let A ∞ denotesthe unique point at infinity of a Zariski closure of φ ( R ) and let ∞ = f − ( A ∞ ) . Thisimplies that g − ( ∞ ) = 1 and we can assume that g − ( ∞ ) = ∞ , i.e., g ∈ R [ t ] . Similarly f i ∈ R [ t ] . Now if deg g = 1 then f : R → R n covers whole φ ( R ) , becausethe image of f is open and closed in φ ( R ) . Otherwise we can compose f with suit-able polynomial of degree two to obtain whole φ ( R ) in the image. In the same waywe can compose f with suitable polynomial of degree one or two to obtain whole ψ ( R ) in the image. In any case ψ ( R ) has a parametrization of degree bounded by deg f ≤ d. (cid:3) Corollary . Let X be a closed semialgebraic set which is R -uniruled andlet f : X → R m be a generically finite polynomial map. Then every connectedcomponent of the set S f is unbounded.
5. An application of the real field case
As an application we give a real counterpart of Theorem 3.1 from [ ]. Theorem . Let G be a real, non-trivial, connected, unipotent group, whichacts effectively and polynomially on a closed semialgebraic R -uniruled set X . Thenthe set F ix ( G ) , of fixed points of G , is also R -uniruled. In particular it does notcontain isolated points. Proof.
First of all let us recall that a connected unipotent group has a normalseries G ⊂ G ⊂ · · · ⊂ G r = G, where G i /G i − ∼ = G a = ( R , + , . By the induction on dim G we can easily reducethe problem to G = G a . Indeed, assume that theorem holds for G = G a . Take aunipotent group G of dim G = n and assume that theorem holds in dimension n − . There is a normal subgroup G n − of dimension n − , such that G/G n − = G a . Moreover, the set R := F ix ( G n − ) is R -uniruled by our hypothesis. Consider the EGREE OF K -UNIRULEDNESS OF S f induced action of the group G a = G/G n − on R. The set of fixed points of thisaction is R -uniruled and it coincides with F ix ( G ) .Hence assume that G = G a . Let D be degree of R -uniruledness of X. Take apoint a ∈ F ix ( G ) . Let φ : G × X ∋ ( g, x ) → φ ( g, x ) ∈ X be a polynomial actionof G on X. This action also induces a polynomial action of the complexification G c = ( C , +) of G on X c . We will denote this action by φ. Assume that deg g φ ≤ d. By Definition 4.2 it is enough to prove, that there exists a parametric curve S ⊂ F ix ( G ) passing through a of degree bounded by max ( d, D ) . Let L be a parametriccurve in X passing through a. If it is contained in
F ix ( G ) , then the assertion istrue. Otherwise consider a closed semialgebraic surface Y = L × G. There is anatural G action on Y : for h ∈ G and y = ( l, g ) ∈ Y we put h ( y ) = ( l, hg ) ∈ Y. Take a map
Φ : L × G ∋ ( x, g ) → φ ( g, x ) ∈ X. It is a generically finite polynomial map. Observe that it is G -invariant, whichmeans Φ( gy ) = g Φ( y ) . This implies that the set S Φ of points at which the map Φ is not finite is G -invariant. Indeed, it is enough to show that the complement ofthis set is G -invariant. Let Φ be finite at x ∈ X. This means that there is an openneighborhood U of x such that the map Φ : Φ − ( U ) → U is finite. Now we havethe following diagram: Φ − ( gU ) = g Φ − ( U )Φ − ( U ) U gU ΦΦ gg ❄ ✲✲ ❄ This shows that the map Φ is finite over U , then it is finite over gU. In particular thisimplies that the set S Φ is G -invariant. Let S Φ = S ∪ S ... ∪ S k be a decompositionof S Φ into parametric curves (see the proof of Theorem 4.9). Since the set S Φ is G -invariant, we have that each parametric curve S i is also G -invariant. Note thatthe point a belongs to S Φ , because the fiber over a has infinite number of points.We can assume that a ∈ S . Let us note that the point a is also a fixed point for G c . Let x ∈ S , we want to show that x ∈ F ix ( G ) . Indeed, the set S c is also G c invariant and if x F ix ( G ) then G c .x = S c and a would be in the orbit of x , whichis a contradiction. Hence S ⊂ F ix ( G ) and we conclude by Theorem 4.5. (cid:3) Corollary . Let G be a real, non-trivial, connected, unipotent group whichacts effectively and polynomially on a closed semialgebraic set X . If the set F ix ( G ) ,of fixed points of G , is nowhere dense in X , then it is R -uniruled. Corollary . Let G be a real, non-trivial connected unipotent group whichacts effectively and polynomially on a connected Nash submanifold X ⊂ R n . Thenthe set F ix ( G ) is R -uniruled. References [1] Z. Jelonek, The set of points at which the polynomial mapping is not proper, Ann. Polon.Math. 58 (1993), 259-266.[2] Z. Jelonek, Testing sets for properness of polynomial mappings, Math. Ann. 315 (1999), 1-35.[3] Z. Jelonek, Topological characterization of finite mappings, Bull. Acad. Polon. Sci. Math. 49(2001), 279-283.[4] Z. Jelonek, Geometry of real polynomial mappings, Math. Z. 239 (2002), 321-333.[5] Z. Jelonek, K. Kurdyka, On asymptotic critical values of a complex polynomial, J. ReineAngew. Math. 565 (2003), 1-11.[6] Z. Jelonek, On the effective Nullstellensatz, Invent. Math. 162 (2005), 1-17.[7] Z. Jelonek, M. Lasoń, The set of fixed points of a unipotent group, J. Algebra 322 (2009),2180-2185.[8] Z. Jelonek, On the Russell problem, J. Algebra 324 (2010), no. 12, 3666-3676.[9] M. Safey El Din, Testing Sign Conditions on a Multivariate Polynomial and Applications,Mathematics in Computer Science 1 (2007), 177-207.[10] A. Stasica, Geometry of the Jelonek set, J. Pure Appl. Algebra 198 (2005), 317-327.[11] H. Vui, P. Són, Representations of positive polynomials and optimization on noncompactsemialgebraic sets, SIAM J. Optim. 20 (2010), 3082-3103.(Z. Jelonek)
Institute of Mathematics of the Polish Academy of Sciences, ul.Śniadeckich8, 00-656 Warszawa, Poland
E-mail address : [email protected] (M. Lasoń) Institute of Mathematics of the Polish Academy of Sciences, ul.Śniadeckich8, 00-656 Warszawa, Poland
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