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 coneof all PQFs $Q$ such that the Delaunay tilings of $\Bbb{Z}^d$ withrespect to the forms $A$ and $Q$ have the same edge structure. Fornon-generic PQFs, the iso edge domain can be defined using allcentrally-symmetric Delaunay polytopes in addition to edges. Thesubdivision of the cone of all $d$-dimensional PQFs into iso edgedomains is a coarsening of the subdivision into secondary cones ofDelaunay tilings, [see M. Dutour-Sikirić et al., ActaCrystallogr. Sect. A {\bf 72} (2016), no.~6, 673--683; MR3573502].\par The authors of this paper review general properties of iso edgedomains and prove that for every dimension, the rationalcompactification of the subdivision of the cone of PQFs results in aprojective compactification (Theorem 3.2). They also prove a generalformula connecting dimensions and stabilizers of iso edge domainscontaining a PQF, the {\it Mass Formula} in Theorem 4.1.\par In addition, the authors compute the decomposition of the cone of5-dimensional PQFs into iso edge domains, with appropriate action of${\rm GL}_5(\Bbb{Z})$, and prove the Conway@-Sloane conjecture onvonorms [J.~H. Conway and N.~J.~A. Sloane, Proc. Roy. Soc. LondonSer. A {\bf 436} (1992), no.~1896, 55--68; MR1177121]for $d=5$.\par The authors also use the computational result to formulate aconjecture that the matroidal locus within the cone of PQFs can becharacterized by positivity of conorms (Conjecture 2), and verify itfor dimensions up to 5.