长割线半弧
Semiarcs with long secants
摘要：
在一个$q$阶的射影平面$Π_q$中，一个非空点集${\cal S}_t$是一个$t$ -半弧，如果在它的每一点上与${\cal S}_t$相切的切线的个数是$t$。如果${\cal S}_t$是$Π_q$，$t<q$中的$t$ -半弧，则每条线与${\cal S}_t$相交的点至多为$q+1-t$。Dover证明了包含$q$共线点的半椭圆（具有$t=1$的半弧）存在于$Π_q$中的唯一条件是$q<3$ .我们证明了，如果$t>1$，那么$t$ -半弧与$q+1-t$共线点存在，只有当$t\geq \sqrt{q-1}$。在$\mathrm{PG}(2,q)$中，我们证明了下界$t\geq(q-1)/2$，只有当${\cal S}_t$是大小为$3(q+1)/2$的Rédei型阻塞集时才相等。我们称两条线的对称差为$V_t$ -配置，其中$t$进一步从每条线移除点。给出了$t$ -半弧包含$V_t$ -构形的条件，并给出了$\mathrm{PG}(2,q)$中$t$ -半弧的完整刻画.
英文摘要：
In a projective plane $Π_q$ of order $q$, a non-empty point set ${\cal S}_t$ is a $t$-semiarc if the number of tangent lines to ${\cal S}_t$ at each of its points is $t$. If ${\cal S}_t$ is a $t$-semiarc in $Π_q$, $t<q$, then each line intersects ${\cal S}_t$ in at most $q+1-t$ points. Dover proved that semiovals (semiarcs with $t=1$) containing $q$ collinear points exist in $Π_q$ only if $q<3$. We show that if $t>1$, then $t$-semiarcs with $q+1-t$ collinear points exist only if $t\geq \sqrt{q-1}$. In $\mathrm{PG}(2,q)$ we prove the lower bound $t\geq(q-1)/2$, with equality only if ${\cal S}_t$ is a blocking set of Rédei type of size $3(q+1)/2$. We call the symmetric difference of two lines, with $t$ further points removed from each line, a $V_t$-configuration. We give conditions ensuring a $t$-semiarc to contain a $V_t$-configuration and give the complete characterization of such $t$-semiarcs in $\mathrm{PG}(2,q)$.
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