We study incidences between points and (constant-degree algebraic) curves in three dimensions, taken from a family $\mathcal{C}$ of curves that have almost two degrees of freedom, meaning that (i) every pair of curves of $\mathcal{C}$ intersect in $O(1)$ points, (ii) for any pair of points p, q, there are only $O(1)$ curves of $\mathcal{C}$ that pass through both points, and (iii) there exists a 6-variate real polynomial F of constant degree, so that a pair p, q of points admit a curve of $\mathcal{C}$ that passes through both of them if and only if $F(p,q)=0$. (As an example, the family of unit circles in ${\mathbb{R}}^3$ that pass through some fixed point is such a family.)

We begin by studying two specific instances of this scenario. The first instance deals with the case of unit circles in ${\mathbb{R}}^3$ that pass through some fixed point (so called anchored unit circles). In the second case we consider tangencies between directed points and circles in the plane, where a directed point is a pair $(p,u)$, where p is a point in the plane and u is a direction, and $(p,u)$ is tangent to a circle γ if $p\in \gamma$ and u is the direction of the tangent to γ at p. A lifting transformation due to Ellenberg et al. [3] maps these tangencies to incidences between points and curves (‘lifted circles’) in three dimensions. In both instances we have a family of curves in ${\mathbb{R}}^3$ with almost two degrees of freedom.

We show that the number of incidences between m points and n anchored unit circles in ${\mathbb{R}}^3$, as well as the number of tangencies between m directed points and n arbitrary circles in the plane, is $O(m^{3/5}{n}^{3/5}+m+n)$.

We then derive a similar incidence bound, with a few additional terms, for more general families $\mathcal{C}$ of curves in ${\mathbb{R}}^3$ with almost two degrees of freedom, under a few additional natural assumptions.

The proofs follow standard techniques, based on polynomial partitioning, but they face a critical novel issue involving the analysis of surfaces that are infinitely ruled by the respective family of curves, as well as of surfaces in a dual three-dimensional space that are infinitely ruled by the respective family of suitably defined dual curves. We either show that no such surfaces exist, or develop and adapt techniques for handling incidences on such surfaces.

The general bound that we obtain is$I(P,C)=O(m^{3/5}{n}^{3/5}+(m^{11/15}{n}^{2/5}+n^{8/9}){\delta }^{1/3}+m^{2/3}{n}^{1/3}{\pi }^{1/3}+m+n),$ where π (resp., δ) is the maximal number of curves (resp., dual curves) that lie on a common surface that is infinitely ruled by the family of these curves.

我们研究了三个维度的点和（恒度代数）曲线之间的发生率，取自一个家庭 $\mathcal{C}$具有几乎两个自由度的曲线，这意味着 (i) 每对曲线$\mathcal{C}$ 相交 $○(1)$点，（ii）对于任何一对点p，q，只有$○(1)$ 的曲线 $\mathcal{C}$通过两个点，并且 (iii) 存在一个恒定次数的 6 变量实数多项式F，因此一对p , q点允许一条曲线为$\mathcal{C}$ 当且仅当 $F(p,q)=0$. （例如，单位圆族${\mathbb{R}}^3$ 通过某个固定点的就是这样一个家庭。）

我们首先研究这个场景的两个具体实例。第一个例子涉及单位圆的情况${\mathbb{R}}^3$通过某个固定点（所谓的锚定单位圆）。在第二种情况下，我们考虑之间相切引导点和圆的平面上，其中有向点是一对$(p,你)$，其中p是平面中的一个点，u是一个方向，并且$(p,你)$与圆γ相切，如果$p\in \gamma$并且u是在p处与γ相切的方向。由于 Ellenberg 等人的提升转换。[3] 将这些切线映射到三个维度上的点和曲线（“提升的圆”）之间的发生率。在这两种情况下，我们都有一系列曲线${\mathbb{R}}^3$ 几乎有两个自由度。

我们证明了m个点和n 个锚定单位圆之间的发生次数${\mathbb{R}}^3$，以及平面中m 个有向点与n 个任意圆之间的切线数，为$○({米}^{3/5}{n}^{3/5}+米+n)$.

然后，我们为更一般的家庭推导出类似的发生率界限，并添加一些附加项 $\mathcal{C}$ 在曲线 ${\mathbb{R}}^3$ 在一些额外的自然假设下，几乎有两个自由度。

证明遵循基于多项式划分的标准技术，但它们面临一个关键的新问题，涉及分析由各自曲线族无限统治的表面，以及无限统治的双三维空间中的表面由相应的适当定义的对偶曲线族。我们要么证明不存在这样的表面，要么开发和调整处理此类表面上的入射的技术。

我们得到的一般界限是$\mathrm{一世}(磷,C)=○({米}^{3/5}{n}^{3/5}+({米}^{11/15}{n}^{2/5}+n^{8/9}){\delta }^{1/3}+{米}^{2/3}{n}^{1/3}{\pi }^{1/3}+米+n),$其中π (resp., δ ) 是位于由这些曲线族无限支配的公共曲面上的曲线（分别为对偶曲线）的最大数量。