Translational and great Darboux cyclides

Lubbes, Niels

Comptes Rendus. Mathématique, Tome 362 (2024) no. G4, pp. 413-448

A surface that is the pointwise sum of circles in Euclidean space is either coplanar or contains no more than 2 circles through a general point. A surface that is the pointwise product of circles in the unit-quaternions contains either 2, 3, 4, or 5 circles through a general point. A surface in a unit-sphere of any dimension that contains 2 great circles through a general point contains either 4, 5, 6, or infinitely many circles through a general point. These are some corollaries from our classification of translational and great Darboux cyclides. We use the combinatorics associated to the set of low degree curves on such surfaces modulo numerical equivalence.

Une surface qui est la somme ponctuelle de cercles dans l’espace euclidien est soit coplanaire, soit ne contient pas plus de 2 cercles passant par un point général. Une surface qui est le produit ponctuel de cercles dans les quaternions unitaires contient soit 2, 3, 4, ou 5 cercles passant par un point général. Une surface dans une sphère unitaire de n’importe quelle dimension qui contient 2 grands cercles passant par un point général contient soit 4, 5, 6, ou une infinité de cercles passant par un point général. Ce sont quelques corollaires de notre classification des cyclides de translation et des cyclides de Darboux. Nous utilisons la combinatoire associée à l’ensemble des courbes de faible degré sur de telles surfaces modulo l’équivalence numérique.

Classification : 51B10, 51M15, 14J17, 14C20
Keywords: real surfaces, pencils of circles, singular locus, Darboux cyclides, Clifford torus, Möbius geometry, elliptic geometry, hyperbolic geometry, Euclidean geometry, Euclidean translations, Clifford translations, unit quaternions, weak del Pezzo surfaces, divisor classes, Néron–Severi lattice
Mots-clés : surfaces réelles, faisceaux de cercles, lieu singulier, cyclides de Darboux, tore de Clifford, géométrie de Möbius, géométrie elliptique, géométrie hyperbolique, géométrie euclidienne, translations euclidiennes, translations de Clifford, quaternions unitaires, surfaces de del Pezzo faibles, classes de diviseurs, réseau de Néron–Severi

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