부산대학교 도서관

This paper studies iso edge domains, also known as $C$-type domains, associated with positive definite quadratic forms (PQFs) or, equivalently, $d$-dimensional lattices. \par For a given (generic) PQF $A$, the iso edge domain of $A$ is the cone of all PQFs $Q$ such that the Delaunay tilings of $\Bbb{Z}^d$ with respect to the forms $A$ and $Q$ have the same edge structure. For non-generic PQFs, the iso edge domain can be defined using all centrally-symmetric Delaunay polytopes in addition to edges. The subdivision of the cone of all $d$-dimensional PQFs into iso edge domains is a coarsening of the subdivision into secondary cones of Delaunay tilings, [see M. Dutour-Sikirić et al., Acta Crystallogr. Sect. A {\bf 72} (2016), no.~6, 673--683; MR3573502]. \par The authors of this paper review general properties of iso edge domains and prove that for every dimension, the rational compactification of the subdivision of the cone of PQFs results in a projective compactification (Theorem 3.2). They also prove a general formula connecting dimensions and stabilizers of iso edge domains containing a PQF, the {\it Mass Formula} in Theorem 4.1. \par In addition, the authors compute the decomposition of the cone of 5-dimensional PQFs into iso edge domains, with appropriate action of ${\rm GL}_5(\Bbb{Z})$, and prove the Conway@-Sloane conjecture on vonorms [J.~H. Conway and N.~J.~A. Sloane, Proc. Roy. Soc. London Ser. A {\bf 436} (1992), no.~1896, 55--68; MR1177121] for $d=5$. \par The authors also use the computational result to formulate a conjecture that the matroidal locus within the cone of PQFs can be characterized by positivity of conorms (Conjecture 2), and verify it for dimensions up to 5.