Incidences Between Points and Lines in ${\mathbb {R}}^4$

Abstract

We show that the number of incidences between m distinct points and n distinct lines in ${\mathbb {R}}^4$ is $O(2^{c\sqrt{\log m}} (m^{2/5}n^{4/5}+m) + m^{1/2}n^{1/2}q^{1/4} + m^{2/3}n^{1/3}s^{1/3} + n)$, for a suitable absolute constant c, provided that no 2-plane contains more than s input lines, and no hyperplane or quadric contains more than q lines. The bound holds without the factor $2^{c\sqrt{\log m}}$ when $m \le n^{6/7}$ or $m \ge n^{5/3}$. Except for the factor $2^{c\sqrt{\log m}}$, the bound is tight in the worst case.

Appendix: Severi’s Theorem

In this appendix we sketch a proof of Severi’s theorem (Theorem 3.11).

First, recall from Sect. 2.1 that a real (resp., complex) surface X is ruled by real (resp., complex) lines if every point $p\in X$ in a Zariski open dense set is incident to a real (resp., complex) line that is fully contained in X. This definition has been used in several recent works (see, e.g.,  [15]); this is a slightly weaker condition than the classical condition that requires that every point of X be incident to a line contained in X. Nevertheless, as we show next, the two are equivalent.

Lemma 6.1

Let $f \in {\mathbb {R}}[x,y,z]\,($resp., $f \in {\mathbb {R}}[x,y,z,w])$ be an irreducible polynomial such that there exists a Zariski open dense set $U\subseteq Z(f)$, so that each point in the set is incident to a line, fully contained in Z(f). Then $\mathsf {FL}_{f}\ ($resp., $\mathsf {FL}_{f}^4)$ vanishes identically on Z(f), and Z(f) is ruled by lines.

Proof

By assumption and definition, $\mathsf {FL}_{f}$ (resp., $\mathsf {FL}_{f}^4$) vanishes on U. If it vanishes on Z(f), Theorem 2.11 implies that Z(f) is ruled. Otherwise, $Z(f, \mathsf {FL}_{f})$ (resp., $Z(f,\mathsf {FL}_{f}^4)$) is properly contained in Z(f) and contains U. Since Z(f) is irreducible, this latter variety must be of dimension at most 1 (resp., 2). On the other hand, $Z(f, \mathsf {FL}_{f})$ (resp., $Z(f,\mathsf {FL}_{f}^4)$) is Zariski closed set (by definition of the Zariski topology) and therefore contains its Zariski closure. As U is Zariski dense, its Zariski closure is Z(f). $\square$

Remark

In Sharir and Solomon [37], we have proved the same statement without using the Flecnode polynomial.

This phenomenon generalizes to k-flats instead of lines (and the proof translates verbatim).

Lemma 6.2

Let V be an irreducible variety for which there exists a Zariski open subset $U\subseteq V$ with the property that each point $p\in U$ is incident to a k-flat that is fully contained in V. Then this property holds for every point of V.

We now proceed to sketch a proof of Severi’s theorem. For convenience, we repeat its statement.

Theorem 3.11

(Severi’s Theorem [34]) Let $X \subset {\mathbb {P}}^d({\mathbb {C}})$ be a k-dimensional irreducible variety, and let $\Sigma _0$ be a component of maximal dimension of F(X), such that the lines of $\Sigma _0$ cover X. Then the following holds.

• 1. If $\dim (\Sigma _0) = 2k - 2$, then X is a copy of $\mathbb {P}^k({\mathbb {C}})\,($that is, a complex projective k-flat).

• 2. If $\dim (\Sigma _0) = 2k -3$, then either X is a quadric, or X is ruled by copies of ${\mathbb {P}}^{k-1}({\mathbb {C}})$, i.e., every point $p\in X$ is incident to a copy of ${\mathbb {P}}^{k-1}({\mathbb {C}})$ that is fully contained in X.

We sketch a proof in the case $k=3,d=4$, under the simplifying assumption that for any non-singular $x\in X,\Sigma _{0,x}$ is infinite; this assumption holds in our application of the theorem (by the informal dimensionality argument mentioned in the paper, it holds “on average” in general for these parameters). Our proof is based on a sketch provided by A. J. de Jong, via private communication, and we are very grateful for his assistance.

Sketch of Proof

For $x \in X$, we recall that $\Xi _{0,x}$ denotes the cone of lines (i.e., union of lines) of $\Sigma _{0,x}$ The proof consists of the following steps.

• (1) Assume first that $\dim (\Sigma _0)=2k-2=4$. Then there exists some non-singular point $x_0\in X$ with $\dim (\Sigma _{0,x_0})=2$. Indeed, if, for all non-singular points $x\in X,\dim (\Sigma _{0,x}) \le 1$, then $\dim (\Sigma _0) < 4$ (see the analysis in Theorem 3.9, and the preceding analysis), contradicting the assumption in this case. By an argument that has already been sketched earlier, this implies that $\dim (\Xi _{0, x_0})=3$, i.e., the cone of lines in $\Sigma _{0,x_0}$ through $x_0$ is three-dimensional, and therefore $X=\Xi _{0,x_0}$. As $x_0$ is non-singular, it follows that X must be a hyperplane, as claimed.

• (2) Consider next the case where $\dim (\Sigma _0)=2k-3=3$, and for any non-singular point $x\in X,\Sigma _{0,x}$ is 1-dimensional (as just argued, if $\Sigma _{0,x}$ is two-dimensional for some non-singular $x \in X$, then X is a hyperplane). In other words, $\Sigma _{0,x}$, parameterized by the direction of its lines, is a curve in ${\mathbb {P}} T_x X \cong {\mathbb {P}}^2({\mathbb {C}})$; put $e_x$ for its degree. If $e_x = 1$, then $\Xi _{0,x}$ contains a 2-flat.

We next define a “plane-flecnode polynomial system” associated with X, that expresses, for a point $x\in X$, the existence of a 2-flat H, such that H osculates to X to order 3 at x. Since X is a hypersurface, we can write $X=Z(f)$, for a suitable 4-variate polynomial f (see Sect. 2), and assume that f is irreducible (as X is irreducible).

We represent a 2-flat through the origin in ${\mathbb {C}}^4$ (ignoring the lower-dimensional family of 2-flats that cannot be represented in this manner) as

$$\begin{aligned} H_{v_0,v_1,v_2,v_3}:= \{(x,y,z,w) \mid z=v_0x+v_1y, \ w=v_2x+v_3y\}, \end{aligned}$$

(46)

for $v_0,v_1,v_2,v_3 \in {\mathbb {C}}$. The 2-flat $H_{v_0,v_1,v_2,v_3}$ is said to osculate to $X=Z(f)$ to order k at p, if the Taylor expansion of f at p along H satisfies

$$\begin{aligned} f(p+(x,y,v_0x+v_1y, v_2x+v_3y))=O(x^{k+1}+y^{k+1}). \end{aligned}$$

(47)

This translates into a system of homogeneous polynomial equations in $v_0,v_1,v_2,v_3$, involving the partial derivatives of f up to order k. Specializing to the case $k=3$, the plane-flecnode polynomial system, $\mathsf {PFL}_{f}$, associated with f, is obtained by eliminating $v_0,v_1,v_2,v_3$ from these equations (for osculation up to order 3). This is the multipolynomial resultant system of the polynomials defining these equations up to order 3, with respect to $v_0,v_1,v_2,v_3$ (see Van der Waerden [45, Chap. XI] for details).

Another theorem of Landsberg [25, Thm. 1] states that, if, for every $g\in \mathsf {PFL}_{f},g$ vanishes identically on X, then X is ruled by 2-flats, which finishes the proof in this case.

Therefore, we may assume that $X\cap Z(\mathsf {PFL}_{f})$ is a Zariski closed proper subset of X. By definition of $\mathsf {PFL}_{f}$, it follows that for every non-singular point $x\in X \setminus Z(\mathsf {PFL}_{f})$ (namely, outside the Zariski closed set $Z(\mathsf {PFL}_{f})$), we have $e_x>1$. Indeed, if $e_x=1$, then, as observed above, there is a 2-flat incident to x, and fully contained in X, implying that for every $g \in \mathsf {PFL}_{f},g(x)=0$, contradicting the assumption that $x \in X \setminus Z(\mathsf {PFL}_{f})$.

For a generic hyperplane H in ${\mathbb {P}}^4({\mathbb {C}})$, which is not contained in $Z(\mathsf {PFL}_{f})$, put $S_H := X \cap H$. As observed above, $X \cap Z(\mathsf {PFL}_{f})$ is properly contained in X, which in turn implies that, for a generic hyperplane H in ${\mathbb {P}}^4({\mathbb {C}}),S_H$ is not fully contained in $Z(\mathsf {PFL}_{f})$. Indeed, let g be a polynomial in $\mathsf {PFL}_{f}$ that does not vanish identically on X. Then $X \cap Z(g) = Z(f,g)$ is strictly contained in $X=Z(f)$, and since Z(f) is irreducible, it follows that Z(f, g) is two-dimensional. Therefore, for a generic hyperplane $H,X \cap Z(\mathsf {PFL}_{f}) \cap H$ is contained in the one-dimensional variety $Z(f,g)\cap H$, and thus cannot contain the two-dimensional variety $S_H$.

Let $x \in X$ be a non-singular point, and let H be a hyperplane in ${\mathbb {P}}^4({\mathbb {C}})$, which is incident to X and not contained in $Z(\mathsf {PFL}_{f})$. We claim that for a generic H, there are $e_x$ distinct lines that are incident to x and fully contained in $S_H$. Indeed, the intersection of the hyperplane H with $T_x X$ is a 2-flat in $T_x X$ containing x. Taking its projectivization (where the point x is regarded as 0), namely, ${\mathbb {P}} T_x X \cong {\mathbb {P}}^2$, the (generic) 2-flat $T_x X \cap H$ becomes a (generic) line. The degree of $\Sigma _{0,x} \subset {\mathbb {P}} T_x X$ is $e_x$. Therefore, the intersection of $\Sigma _x$ with a line in ${\mathbb {P}} T_x X\cong {\mathbb {P}}^2$ consists of $e_x$ points, which are distinct since the line is generic. Therefore, its intersection with $\Sigma _{0,x}$ consists of $e_x$ distinct points. These $e_x$ distinct (projective) points represent $e_x$ distinct lines, incident to x and fully contained in $X\cap H=S_H$, as claimed.

We say that a pair (x, H), where H is a hyperplane in ${\mathbb {P}}^4({\mathbb {C}})$ and $x \in S_H$, is adequate if there are $e_x$ distinct lines incident to x that are fully contained in $S_H$. Since a generic point x is non-singular, the previous paragraph implies that a generic pair (x, H) is adequate. Therefore, by changing the order of quantifiers, fixing a generic hyperplane H, a generic point $x \in S_H$ is such that the pair (x, H) is adequate.

By Bertini’s Theorem (see, e.g., Harris [17, Thm. 17.16]), the irreducibility of X implies that for a generic hyperplane H, the surface $S_H$ is an irreducible surface in $H\cong \mathbb {P}^3({\mathbb {C}})$. For a generic point $x \in S_H$, that is, outside an algebraic curve ${\mathcal {C}}_H$ in $S_H$, the pair (x, H) is adequate. Therefore, there are $e_x$ distinct lines that are incident to x and fully contained in $S_H$, which, by Lemma 6.1, implies that $S_H$ is a ruled surface. Moreover, for any $x \in S_H\setminus Z(\mathsf {PFL}_{f})$, we have $e_x>1$. As observed above, $\mathsf {PFL}_{f}$ does not vanish identically on $S_H$, implying that $Z(\mathsf {PFL}_{f})\cap S_H$ is a Zariski closed proper subset of $S_H$, i.e., an algebraic curve contained in $S_H$. Adding this curve to ${\mathcal {C}}_H$, it follows that outside this algebraic curve, each point of $S_H$ is incident to at least two lines fully contained in $S_H$. By Sharir and Solomon [37, Lem. 9], this implies that $S_H$ is either a 2-flat or a regulus. If X is of degree greater than two, then, for a generic hyperplane $H,S_H$ is a (two-dimensional) surface of degree greater than two. Therefore, X must be of degree at most two, namely, X is either a hyperplane or a quadric. If X is a hyperplane, then $\Sigma _{0}$ is four-dimensional, contrary to the present assumption, so finally, we deduce that X is a quadric, and the proof is complete. $\square$