No Arabic abstract
We study SUSY $N$-supergroups, $N=1,2$, their classification and explicit realization, together with their real forms. In the end, we give the supergroup of SUSY preserving automorphism of $mathbf{C}^{1|1}$ and we identify it with a subsupergroup of the SUSY preserving automorphisms of $mathbf{P}^{1|1}$.
The intersection of a quadric and a cubic surface in 3-space is a canonical curve of genus 4. It has 120 complex tritangent planes. We present algorithms for computing real tritangents, and we study the associated discriminants. We focus on space sextics that arise from del Pezzo surfaces of degree one. Their numbers of planes that are tangent at three real points vary widely; both 0 and 120 are attained. This solves a problem suggested by Arnold Emch in 1928.
We prove the failure of the local-global principle, with respect to all discrete valuations, for isotropy of quadratic forms over a rational function field of transcendence degree at least 2 over the complex numbers. Our construction involves the generalized Kummer varieties considered by Borcea and Cynk--Hulek.
In this paper we prove that the etale sheafification of the functor arising from the quotient of an algebraic supergroup by a closed subsupergroup is representable by a smooth superscheme.
We give conditions for unitarizability of Harish-Chandra super modules for Lie supergroups and superalgebras.
We construct smooth rational real algebraic varieties of every dimension $ge$ 4 which admit infinitely many pairwise non-isomorphic real forms.