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

TL;DR

This paper extends previous results on the structure of the non-properness set of polynomial maps from real and complex fields to positive characteristic fields, providing a geometric proof of an upper bound on its degree.

Contribution

It generalizes the known properties of the non-properness set to positive characteristic fields and offers a geometric proof for the degree bound.

Findings

The non-properness set is covered by polynomial curves of degree at most d.

The degree bound for the non-properness set is generalized to positive characteristic.

A geometric proof of the degree bound is provided.

Abstract

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 $d$.

Equations

$$\frac{\mathrm{deg}X{\prod }_i\mathrm{deg}{f}_i-\mu (f)}{{\text{min}}_i\mathrm{deg}{f}_i},$$

$$\mathrm{graph}(f)=\{(x,y)\in X\times Y:y_i=f_i(x)\},$$

$$S_f=\pi (\overline{\mathrm{graph}(f)}\setminus \mathrm{graph}(f)),$$

$$A_c:=A\cap \{x\in {\mathbb{K}}^n:f(x)=c\}$$

$${\phi }_{a,b}:\mathbb{K}\ni t\to (a_1+b_{1,1}t+\cdots +b_{1,d}^d{t}^d,\dots ,a_n+b_{n,1}t+\cdots +b_{n,d}^d{t}^d)\in {\mathbb{K}}^n$$

$${\mathbb{K}}^n\times {\mathbb{P}}^{dn-1}\supset V=\{(a,b)\in {\mathbb{K}}^n\times {\mathbb{P}}^{dn-1}:{\forall }_{t,i}{g}_i({\phi }_{a,b}(t))=0\text{and}$$

$${\forall }_{t_1,t_2}f({\phi }_{a,b}(t_1))=f({\phi }_{a,b}(t_2))\}\ni (a,b)\to a\in {\mathbb{K}}^n.$$

$$S_f=\pi (\overline{\mathrm{graph}(f)}\setminus \mathrm{graph}(f)).$$

$$z\in \overline{\mathrm{graph}(f)}\setminus \mathrm{graph}(f)=\overline{\mathrm{graph}(f)}\cap \{x_0=0\}.$$

$$A_c:=A\cap \{x\in {\mathbb{K}}^n:f(x)=c\}=\overline{\mathrm{graph}(f)}\cap \{x_i\ne 0\}\cap \{x_0=c{x}_i\}.$$

$$\mathbb{K}\ni t\to (c:a_1:\cdots :a_{i-1}:1:\cdots :a_{j-1}:t:a_{j+1}:\dots )\stackrel{f}{\to }{A}_c\subset \mathrm{graph}(f).$$

$$A_0=\overline{\mathrm{graph}(f)}\cap \{x_i\ne 0\}\cap \{x_0=0\}$$

$$\overline{\mathrm{graph}(f)}\setminus \mathrm{graph}(f)=\overline{\mathrm{graph}(f)}\cap \{x_0=0\}.$$

Full text

Research supported by Polish National Science Centre grant no. 2013/09/B/ST1/04162.

1. Introduction

We begin by recalling some necessary definitions, notions and facts (cf. [10]). Unless stated otherwise, $\mathbb{K}$ is an arbitrary algebraically closed field. All affine varieties are considered to be embedded in an affine space.

An irreducible affine curve $\Gamma\subset\mathbb{K}^{m}$ is called a parametric curve of degree at most $d$, if there exists a non-constant polynomial map $f:\mathbb{K}\rightarrow\Gamma$ of degree at most $d$ (by degree of $f=(f_{1},\dots,f_{m})$ 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 $X\subset\mathbb{K}^{m}$ be an irreducible affine variety of dimension $n$, and let $d$ be a constant. The following conditions are equivalent:

1. (1)

for every point $x\in X$ there exists a parametric curve $l_{x}\subset X$ of degree at most $d$ passing through $x$, 2. (2)

there exists an open, non-empty subset $U\subset X$, such that for every point $x\in U$ there exists a parametric curve $l_{x}\subset X$ of degree at most $d$ passing through $x$, 3. (3)

there exists an affine variety $W$ of dimension $\dim X-1$, and a dominant polynomial map $\phi:\mathbb{K}\times W\ni(t,w)\to\phi(t,w)\in X$ such that $\deg_{t}\phi\leq d$.

We say that an affine variety $X$ has degree of $\mathbb{K}$-uniruledness at most $d$ if all its irreducible components satisfy the conditions of Proposition 1.1 (cf. Definition 1.3 [10]). An affine variety is called $\mathbb{K}$-uniruled if it has some degree of $\mathbb{K}$-uniruledness. To simplify the notion we assume that the empty set has degree of $\mathbb{K}$-uniruledness equal to zero, in particular it is $\mathbb{K}$-uniruled.

Let $f:X\rightarrow Y$ be a generically finite polynomial map between affine varieties. Recall that $f$ is finite (proper) at a point $y\in Y$, if there exists an open neighborhood $U$ of $y$ such that $f|_{f^{-1}(U)}:f^{-1}(U)\rightarrow U$ is a finite map. 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 $f:X\rightarrow Y$ be a generically finite polynomial map between affine varieties. The set $S_{f}$ is a hypersurface in $\overline{f(X)}$ or it is empty. Additionally, if $X$ is $\mathbb{K}$-uniruled, then the set $S_{f}$ is also $\mathbb{K}$-uniruled.

Theorem 1.3** (Theorem 4.6 [7]).**

Let $f:X\to\mathbb{K}^{n}$ be a dominant, generically finite, separable, polynomial map between affine varieties. Then the set $S_{f}$ is a hypersurface of degree at most

[TABLE]

where $\mu(f)$ is the multiplicity of $f$, or it is empty.

Due to Theorem 1.2, if $X$ is a $\mathbb{K}$-uniruled affine variety and $f:X\rightarrow\mathbb{K}^{m}$ is a generically finite polynomial map, then the set $S_{f}$ is $\mathbb{K}$-uniruled. In [10] we studied the behavior of degree of $\mathbb{K}$-uniruledness of the set $S_{f}$ as a function of degree of the map $f$.

Theorem 1.4** (Theorem 3.2 [10]).**

Let $f:\mathbb{C}^{n}\rightarrow\mathbb{C}^{m}$ be a generically finite polynomial map of degree $d$. Then the set $S_{f}$ has degree of $\mathbb{C}$-uniruledness at most $d-1$.

Theorem 1.5** (Theorem 3.5 [10]).**

Suppose $X$ is an affine variety with degree of $\mathbb{C}$-uniruledness at most $d_{1}$. Let $f:X\rightarrow\mathbb{C}^{m}$ be a generically finite polynomial map of degree $d_{2}$. Then the set $S_{f}$ has degree of $\mathbb{C}$-uniruledness at most $d_{1}d_{2}$.

In [10] we provide examples showing that the above bounds are tight. Their proofs intensively use the topology of $\mathbb{C}$, 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 $\mathbb{K}$ 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 $\mathbb{K}$-uniruledness at most $d$. Our proof uses cute and simple geometric idea.

2. Main result

Proposition 2.1** ([3]).**

Let $f:X\rightarrow Y$ be a generically finite map between affine varieties $X\subset\mathbb{K}^{n}$ and $Y\subset\mathbb{K}^{m}$. Let

[TABLE]

and let $\overline{\operatorname{graph}(f)}$ be its closure in $\mathbb{P}^{n}\times\mathbb{K}^{m}$. Then there is an equality

[TABLE]

where $\pi$ denotes the projection to the second factor $\mathbb{P}^{n}\times\mathbb{K}^{m}\rightarrow\mathbb{K}^{m}$.

Lemma 2.2**.**

Let $A\subset\mathbb{K}^{n}$ be an affine set, and let $f$ be a regular function on $\mathbb{K}^{n}$ not equal to [math] on any component of $A$. Suppose that for each $c\in\mathbb{K}\setminus\{0\}$ the set

[TABLE]

has degree of $\mathbb{K}$-uniruledness at most $d$. Then the set $A_{0}$ also has degree of $\mathbb{K}$-uniruledness at most $d$.

Proof.

Suppose affine set is given by $A=\{x\in\mathbb{K}^{n}:g_{1}(x)=\dots=g_{r}(x)=0\}$. For $a=(a_{1},\dots,a_{n})\in\mathbb{K}^{n}$ and $b=(b_{1,1}:\dots:b_{d,n})\in\mathbb{P}^{dn-1}$, let

[TABLE]

be a parametric curve of degree at most $d$. Let us consider a variety and a projection

[TABLE]

[TABLE]

The definition of the set $V$ says that parametric curves $\varphi_{a,b}$ 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 $c\in\mathbb{K}\setminus\{0\}$, since they are filled with parametric curves of degree at most $d$. But since $\mathbb{P}^{dn-1}$ 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 $d$. ∎

Theorem 2.3**.**

Let $\mathbb{K}$ be an arbitrary algebraically closed field. If $f:\mathbb{K}^{n}\rightarrow\mathbb{K}^{m}$ is a generically finite polynomial map of degree $d$, then the set $S_{f}$ has degree of $\mathbb{K}$-uniruledness at most $d$.

Proof.

If $n=1$, then the map is proper and $S_{f}$ is empty. Suppose $n\geq 2$. Due to Proposition 2.1

[TABLE]

It is enough to prove that the set $\overline{\operatorname{graph}(f)}\setminus\operatorname{graph}(f)$ is filled with parametric curves of degree at most $d$. Indeed, we can take images of these curves under projection $\pi$. 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 $\pi$, 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 $\mathbb{P}^{n}\times\mathbb{K}^{m}$ by $(x_{0}:\dots:x_{n};x_{n+1},\dots,x_{n+m})$. Take an arbitrary point

[TABLE]

We are going to show that through $z$ passes a parametric curve of degree at most $d$. Since $z_{0}=0$, there exists $1\leq i\leq n$ such that $z_{i}\neq 0$. Consider an affine set $A:=\overline{\operatorname{graph}(f)}\cap\{x_{i}\neq 0\}$ and a function $f=\frac{x_{0}}{x_{i}}$ regular on it. Consider sets

[TABLE]

For $c\neq 0$ 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

[TABLE]

Hence, by Lemma 2.2, the set

[TABLE]

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

[TABLE]

This finishes the proof. ∎

By looking carefully at the above proof we get the following slightly more general result.

Corollary 2.4**.**

Let $\mathbb{K}$ be an arbitrary algebraically closed field, and let $W$ be an affine variety. If $f:\mathbb{K}^{2}\times W\rightarrow\mathbb{K}^{m}$ is a generically finite polynomial map of degree $d$, then the set $S_{f}$ has degree of $\mathbb{K}$-uniruledness at most $d$.

If we were able to prove the assertion of the corollary for maps $f:\mathbb{K}\times W\rightarrow\mathbb{K}^{m}$, this would imply Theorem 1.5 for arbitrary algebraically closed fields (see the proof of Theorem 3.5 [10]).

3. Remarks

The gap between characteristic zero (Theorem 1.4) and arbitrary characteristic (Theorem 2.3) suggests the following.

Conjecture 3.1**.**

Let $\mathbb{K}$ be an arbitrary algebraically closed field. If $f:\mathbb{K}^{n}\rightarrow\mathbb{K}^{m}$ is a generically finite polynomial map of degree $d$, then the set $S_{f}$ has degree of $\mathbb{K}$-uniruledness at most $d-1$.

When we consider all generically finite maps $f:\mathbb{K}^{n}\rightarrow\mathbb{K}^{n}$ of degree at most $d$, then by Theorem 1.2 hypersurfaces $S_{f}$ are all $\mathbb{K}$-uniruled, and by Theorem 1.3 degrees of this hypersurfaces are bounded. In Theorem 2.3 we show that their degree of $\mathbb{K}$-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 $\mathbb{K}$-uniruled hypersurface in $\mathbb{K}^{n}$ of degree at most $d$ has degree of $\mathbb{K}$-uniruledness at most $D(n,d)$?

We ask even a stronger question.

Question 3.3**.**

Does for every $n$ and $d$ exist a universal constant $D=D(n,d)$, such that if a hypersurface in $\mathbb{K}^{n}$ of degree at most $d$ contains a parametric curve passing through $O=(0,\dots,0)$, then it also contains such a curve of degree at most $D(n,d)$?

One can show that positive answer to Question 3.3 is equivalent (for uncountable field $\mathbb{K}$) to the fact that in the set of all hypersurfaces in $\mathbb{K}^{n}$ of a bounded degree, the set of $\mathbb{K}$-uniruled hypersurfaces forms a closed subset.