FoCM 2014 conference
Workshop A3 - Computational Number Theory
December 12, 15:10 ~ 15:50 - Room A21
Concurrent lines on del Pezzo surfaces of degree one
Ronald van Luijk
Universiteit Leiden, Netherlands - rmluijk@gmail.com
Let $k$ be a field and $\overline{k}$ an algebraic closure. A del Pezzo surface over $k$ is a surface over $k$ that is isomorphic over $\overline{k}$ to either $\mathbb{P}^1 \times \mathbb{P}^1$ (degree $8$), or $\mathbb{P}^2$ blown up at $r \leq 8$ points in general position (degree $9-r$). Famous examples (with $r=6$ and degree $3$) are smooth cubic surfaces in $\mathbb{P}^3$, which over $\overline{k}$ contain $27$ lines; at most three of these can be concurrent, that is, go through the same point. Analogously, we get $240$ lines for $r=8$ and degree $1$. Based on the graph on these lines, with edges between those that intersect, we get an upper bound of $16$ for the number of concurrent lines. We show that this upper bound is only attained in characteristic $2$, which makes the case $r=8$ different from all other cases. In most characteristics, including characteristic $0$, the upper bound is $10$.
Joint work with Rosa Winter (Universiteit Leiden, Netherlands).