We study real nonsingular cubic hypersurfaces X ⊂ P 5 up to deformation equivalence combined with projective equivalence and prove that they are classified by the conjugacy classes of involutions induced by the complex conjugation in H 4 (X). Moreover, we provide a graph Γ K4 whose vertices represent the equivalence classes of such cubics and whose edges represent their adjacency. It turns out that the graph Γ K4 essentially coincides with the graph Γ K3 characterizing a certain adjacency of real nonpolarized K3-surfaces.The most familiar logics in the modal family are constructed from a weak logic called K (after Saul Kripke). Under the narrow reading, modal logic concerns necessity and possibility. A variety of different systems may be developed for such logics using K as a foundation.Stanford Encyclopedia of Philosophy, Modal Logic.
For any k ≥ 3, I construct infinitely many pairwise smoothly nonisotopic smooth surfaces F ⊂ CP 2 homeomorphic to a non-singular algebraic curve of degree 2k, realizing the same homology class as such a curve and having abelian fundamental group π 1 (CP 2 F ). This gives an answer to Problem 4.110 in the Kirby list [K].
According to our previous results, the conjugacy class of the involution induced by the complex conjugation in the homology of a real non-singular cubic fourfold determines the fourfold up to projective equivalence and deformation. Here, we show how to eliminate the projective equivalence and obtain a pure deformation classification, that is, how to respond to the chirality problem: which cubics are not deformation equivalent to their image under a mirror reflection. We provide an arithmetical criterion of chirality, in terms of the eigen-sublattices of the complex conjugation involution in homology, and show how this criterion can be effectively applied taking as examples M -cubics (that is, those for which the real locus has the richest topology) and (M − 1)-cubics (the next case with respect to complexity of the real locus). It happens that there is one chiral class of M -cubics and three chiral classes of (M − 1)-cubics, in contrast to two achiral classes of M -cubics and three achiral classes of (M − 1)-cubics.L'univers est un ensemble dissymétrique, et je suis persuadé que la vie, telle qu'elle manifesteà nous, est fonction de la dissymétrie de l'univers ou des conséquences qu'elle entraîne.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.