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Abstract. Each Scorza variety and its secant varieties in the ambient projective space are identified, in the realm of singular Poisson-Kähler geometry, in terms of projectivizations of holomorphic nilpotent orbits in suitable Lie algebras of hermitian type, the holomorphic nilpotent orbits, in turn, being affine varieties. The ambient projective space acquires an exotic Kähler structure, the closed stratum being the Scorza variety and the closures of the higher strata its secant varieties. In this fashion, the secant varieties become exotic projective varieties. In the rank 3 case, the four regular Scorza varieties coincide with the four critical Severi varieties. In the standard cases, the Scorza varieties and their secant varieties arise also via Kähler reduction. An interpretation in terms of constrained mechanical systems is included.

One of the most successful approaches to geometry is the one suggested by Felix Klein. According to Klein, a geometry is a G-space M, that is, a set M together with a group G of transformations of M. This approach provides a powerful link between geometry and algebra. Of particular importance is the situation when the group G acts transitively on

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cohomogeneity one

Each of the four critical Severi varieties arises from a minimal holomorphic nilpotent orbit in a simple regular rank 3 hermitian Lie algebra and each such variety lies as singular locus in a cubic—the chordal variety—in the corresponding complex projective space; the cubic and projective space are identified in terms of holomorphic nilpotent orbits. The projective space acquires an exotic Kähler structure with three strata, the cubic is an example of an exotic projective variety with two strata, and the corresponding Severi variety is the closed stratum in the exotic variety as well as in the exotic projective space. In the standard cases, these varieties arise also via Kähler reduction. An interpretation in terms of constrained mechanical systems is included.

After a brief review of string and M-Theory we point out some deficiencies. Partly to cure them, we present several arguments for “F-Theory”, enlarging spacetime to (2,10) signature, following the original suggestion of C. Vafa. We introduce a suggestive Supersymmetric 27-plet of particles, associated to the exceptional symmetric hermitian space E6/Spin c (10). Several possible future directions, including using projective rather than metric geometry, are mentioned. We should emphasize that F-Theory is yet just a very provisional attempt, lacking clear dynamical principles. Keywords: M-theory, F-theory, Euler multiplets