A $(24_4,32_3)$-configuration on the Schur quartic with logarithmic Chern slope $14/5$

Abstract

Let $X\subset {\mathbb{P}}^3$ be the Schur quartic \[ x_0^4-x_0x_1^3-x_2^4+x_2x_3^3=0. \] We exhibit a connected arrangement of $24$ lines on $X$, defined over $\mathbb{Q}(\sqrt{-3})$, whose singular locus consists of $32$ ordinary triple points and no other intersections. Each line contains four triple points. The resulting reduced divisor $D$ satisfies $D\sim 6H$, where $H$ is the hyperplane class. If $\pi :Y\to X$ blows up the triple points and $B=({\pi }^{-1}D{)}_{\mathrm{red}}$, then \[ \overline{c}_{1}^{2}(Y,B)=112,\qquad \overline{c}_{2}(Y,B)=40, \qquad \frac{\overline{c}_{1}^{2}(Y,B)}{\overline{c}_{2}(Y,B)}=\frac{14}{5}. \] This gives a negative answer to the K3-surface specialization of the proposed $8/3$ bound for transversal arrangements of rational curves. The configuration is one half of the $48$ lines of the second kind on $X$; an explicit projective automorphism exchanges the two halves. We deliver the line parametrizations and all $32$ triple-point coordinates. Ancillary exact-arithmetic data record the $120$ line-containment coefficients and all $276$ pair-incidence determinants. A finite-field mixed-integer search is described only as the discovery procedure and is not used in the proof.

17 pages, 1 figure