Quantitative properties of the non-properness set of a polynomial map, a positive characteristic case

Let $f:\mathbb{K}^n\rightarrow\mathbb{K}^m$ be a generically finite polynomial map of degree $d$ between affine spaces. In arXiv:1411.5011 we proved that if $\mathbb{K}$ is the field of complex or real numbers, then the set $S_f$ of points at which $f$ is not proper, is covered by polynomial curves of degree at most $d-1$. In this paper we generalize this result to positive characteristic. We provide a geometric proof of an upper bound by $d$.


Introduction
We begin by recalling some necessary definitions, notions and facts (cf. [10]). Unless stated otherwise, K is an arbitrary algebraically closed field. All affine varieties are considered to be embedded in an affine space.
An irreducible affine curve Γ ⊂ K m is called a parametric curve 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 1 , . . . , f m ) we mean max i deg f i ). A curve is called parametric if it is parametric of some degree. Proposition 1.1 (Proposition 1.2 [10], cf. Proposition 2.4 [9]). 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 at most 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 passing through x, (3) there exists an affine variety W of dimension dim X − 1, and a dominant polynomial map φ : To simplify the notion we assume that the empty set has degree of K-uniruledness equal to zero, in particular it is K-uniruled. Let f : X → Y be a generically finite polynomial map between affine varieties.
The set of points at which f is not finite (proper) we denote by S f (see [1], and also [2,4,12]). The set S f is a good measure of non-properness of the map f , moreover it has interesting applications [5,6,11,13].
Theorem 1.2 (Theorem 4.1 [8]). Let f : X → Y be a generically finite polynomial map between affine varieties. The set S f is a hypersurface in f (X) or it is empty. Additionally, if X is K-uniruled, then the set S f is also K-uniruled. Theorem 1.3 (Theorem 4.6 [7]). Let f : X → K n be a dominant, generically finite, separable, polynomial map between affine varieties. Then the set S f is a hypersurface of degree at most where µ(f ) is the multiplicity of f , or it is empty.
Due to Theorem 1.2, if X is a K-uniruled affine variety and f : X → K m is a generically finite polynomial map, then the set S f is K-uniruled. In [10] we studied the behavior of degree of K-uniruledness of the set S f as a function of degree of the map f .  In [10] we provide examples showing that the above bounds are tight. Their proofs intensively use the topology of C, thus they can not be adapted for an arbitrary algebraically closed field. However, by the Lefschetz principle, both statements are true for an arbitrary algebraically closed field K of characteristic zero.
In this paper we deal with the positive characteristic case. In Theorem 2.3 we almost (up to additive constant 1) generalize Theorem 1.4. That is, we prove that the set S f has degree of K-uniruledness at most d. Our proof uses cute and simple geometric idea.

Main result
). Let f : X → Y be a generically finite map between affine varieties X ⊂ K n and Y ⊂ K m . Let and let graph(f ) be its closure in P n × K m . Then there is an equality where π denotes the projection to the second factor P n × K m → K m . Lemma 2.2. Let A ⊂ K n be an affine set, and let f be a regular function on K n not equal to 0 on any component of A. Suppose that for each c ∈ K \ {0} the set A c := A ∩ {x ∈ K n : f (x) = c} has degree of K-uniruledness at most d. Then the set A 0 also has degree of Kuniruledness at most d.
Proof. Suppose affine set is given by A = {x ∈ K n : g 1 (x) = · · · = g r (x) = 0}. For a = (a 1 , . . . , a n ) ∈ K n and b = (b 1,1 : · · · : b d,n ) ∈ P dn−1 , let . . , a n + b n,1 t + · · · + b d n,d t d ) ∈ K n be a parametric curve of degree at most d. Let us consider a variety and a projection The definition of the set V says that parametric curves ϕ a,b are contained in A and f is constant on them. Hence V is closed and the image of the projection is contained in A. Moreover the image contains every A c for c ∈ K \ {0}, since they are filled with parametric curves of degree at most d. But since P dn−1 is complete and V is closed, the image of the projection is closed. Hence it must be the whole set A. In particular A 0 is contained in the image, so it is filled with parametric curves of degree at most d. Proof. If n = 1, then the map is proper and S f is empty. Suppose n ≥ 2. Due to Proposition 2.1 It is enough to prove that the set graph(f ) \ graph(f ) is filled with parametric curves of degree at most d. Indeed, we can take images of these curves under projection π. Projection has degree 1, so images of these curves are either parametric curves of degree at most d, or points. Since map f , and as a consequence also π, is generically finite, only on a codimension 1 subvariety images of curves will become points. This by Proposition 1.1 is acceptable.
Let us denote coordinates in P n × K m by (x 0 : · · · : x n ; x n+1 , . . . , x n+m ). Take an arbitrary point We are going to show that through z passes a parametric curve of degree at most d. Since z 0 = 0, there exists 1 ≤ i ≤ n such that z i = 0. Consider an affine set A := graph(f ) ∩ {x i = 0} and a function f = x0 xi regular on it. Consider sets For c = 0 sets A c are filled with parametric curves of degree at most d. Indeed, we can take an index j distinct from 0, i and consider curves K ∋ t → (c : a 1 : · · · : a i−1 : 1 : · · · : a j−1 : t : a j+1 : . . . )

Hence, by Lemma 2.2, the set
is also filled with such curves. In particular we get that though z passes a parametric curve of degree at most d which is contained in This finishes the proof.
By looking carefully at the above proof we get the following slightly more general result.
Corollary 2.4. Let K be an arbitrary algebraically closed field, and let W be an affine variety. If f : K 2 × W → K m is a generically finite polynomial map of degree d, then the set S f has degree of K-uniruledness at most d.
If we were able to prove the assertion of the corollary for maps f : K×W → K m , this would imply Theorem 1.5 for arbitrary algebraically closed fields (see the proof of Theorem 3.5 [10]).

Remarks
The gap between characteristic zero (Theorem 1.4) and arbitrary characteristic (Theorem 2.3) suggests the following. When we consider all generically finite maps f : K n → K n of degree at most d, then by Theorem 1.2 hypersurfaces S f are all K-uniruled, and by Theorem 1.3 degrees of this hypersurfaces are bounded. In Theorem 2.3 we show that their degree of K-uniruledness is also bounded (by d). It is reasonable to ask the following general question.
Question 3.2. Does for every n and d exist a universal constant D = D(n, d), such that evey K-uniruled hypersurface in K n of degree at most d has degree of K-uniruledness at most D(n, d)?
We ask even a stronger question. One can show that positive answer to Question 3.3 is equivalent (for uncountable field K) to the fact that in the set of all hypersurfaces in K n of a bounded degree, the set of K-uniruled hypersurfaces forms a closed subset.