论文标题
在3个空间中有限制线的富裕点和发病率上
On rich points and incidences with restricted sets of lines in 3-space
论文作者
论文摘要
让$ l $为$ r^3 $中的$ n $行，当在$ r^3 $的四维plücker线路中表示为$ r^3 $的点，在不可约的$ t $ t $ constant学位上，该$ t $ t $ constant n as \ emph {nonthent emph {nontionny emph {nontementementere}，与$ l $有关（见下文）。我们显示： \ Medskip \ noindent {\ bf（1）}如果$ t $是二维的，则$ r $ - r $ -rich点的数量（至少$ r $ $ l $）为$ O（n^{4/3+ε}/r^2）$ $ l $ lies在任何共同的法制上，最多有$ o（n^{4/3+ε}）$ 2 $ -RICH点。对于大于一些足够大的常数的$ r $，$ r $ - 富集点的数量也为$ o（n/r）$。 作为一个应用程序，我们推断出（指数中的$ε$ -LOSS）由Pach和de Zeeuw（2107）获得的界限对由$ n $点确定的不同距离的不同距离的数量，在平面上不可减少的代数曲线，这不是线，也不是圆圈。 \ medskip \ noindent {\ bf（2）}如果$ t $是二维，则$ l $和$ r^3 $ in $ r^3 $ in $ o（m+n）$之间的发病率数。 \ medskip \ noindent {\ bf（3）}如果$ t $是三维且非线性的，则$ l $和$ r^3 $中的$ m $点之间的发病率数为$ o \ as $ o \ weft（m^{3/5} M^{1/3} n^{2/3}）s^{1/3} + m + n \ right）$，前提是没有任何平面包含超过$ s $。当$ s = o（\ min \ {n^{3/5}/m^{2/5}，m^{1/2} \}）$，bound bong of ush of of use of of bose $ o（m^{3/5} n^{3/5} {3/5}+m+n）$。 作为一个应用程序，我们证明$ m $点和$ n $行之间的$ r^4 $中包含的二次超曲面（不包含超平面）的发病率数为$ O（m^{3/5} n^{3/5} {3/5} + m + n）$。 除了代数几何形状的各种工具外，这些证明都使用了平面中点和代数曲线之间的发病率数量的最新界限。
Let $L$ be a set of $n$ lines in $R^3$ that is contained, when represented as points in the four-dimensional Plücker space of lines in $R^3$, in an irreducible variety $T$ of constant degree which is \emph{non-degenerate} with respect to $L$ (see below). We show: \medskip \noindent{\bf (1)} If $T$ is two-dimensional, the number of $r$-rich points (points incident to at least $r$ lines of $L$) is $O(n^{4/3+ε}/r^2)$, for $r \ge 3$ and for any $ε>0$, and, if at most $n^{1/3}$ lines of $L$ lie on any common regulus, there are at most $O(n^{4/3+ε})$ $2$-rich points. For $r$ larger than some sufficiently large constant, the number of $r$-rich points is also $O(n/r)$. As an application, we deduce (with an $ε$-loss in the exponent) the bound obtained by Pach and de Zeeuw (2107) on the number of distinct distances determined by $n$ points on an irreducible algebraic curve of constant degree in the plane that is not a line nor a circle. \medskip \noindent{\bf (2)} If $T$ is two-dimensional, the number of incidences between $L$ and a set of $m$ points in $R^3$ is $O(m+n)$. \medskip \noindent{\bf (3)} If $T$ is three-dimensional and nonlinear, the number of incidences between $L$ and a set of $m$ points in $R^3$ is $O\left(m^{3/5}n^{3/5} + (m^{11/15}n^{2/5} + m^{1/3}n^{2/3})s^{1/3} + m + n \right)$, provided that no plane contains more than $s$ of the points. When $s = O(\min\{n^{3/5}/m^{2/5}, m^{1/2}\})$, the bound becomes $O(m^{3/5}n^{3/5}+m+n)$. As an application, we prove that the number of incidences between $m$ points and $n$ lines in $R^4$ contained in a quadratic hypersurface (which does not contain a hyperplane) is $O(m^{3/5}n^{3/5} + m + n)$. The proofs use, in addition to various tools from algebraic geometry, recent bounds on the number of incidences between points and algebraic curves in the plane.
