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We discuss the geometry of rational maps from a projective space of an arbitrary dimension to the product of projective spaces of lower dimensions induced by linear projections. In particular, we give an algebro-geometric variant of the projective reconstruction theorem by Hartley and Schaffalitzky.
We present ALMA [CII] line and far-infrared (FIR) continuum observations of seven z > 6 low-luminosity quasars (M1450 > −25 mag) discovered by our on-going Subaru Hyper Suprime-Cam survey. The [CII] line was detected in all targets with luminosities of ∼(2−10) × 108 L⊙, about one order of magnitude smaller than optically luminous quasars. Also found was a wide scatter of FIR continuum luminosity, ranging from LFIR < 1011L⊙ to ∼2 × 1012L⊙. With the [CII]-based dynamical mass, we suggest that a significant fraction of low-luminosity quasars are located on or even below the local Magorrian relation, particularly at the massive end of the galaxy mass distribution. This is a clear contrast to the previous finding that luminous quasars tend to have overmassive black holes relative to the relation. Our result is expected to show a less-biased nature of the early co-evolution of black holes and their host galaxies.
We show that the moduli space of parabolic bundles on the projective line and the polygon space are isomorphic, both as complex manifolds and as symplectic manifolds equipped with structures of completely integrable systems, if the stability parameters are small.
We introduce a completely integrable system on the Grassmannian of 2-planes in an n-space associated with any triangulation of a polygon with n sides, and we compute the potential function for its Lagrangian torus fiber. The moment polytopes of this system for different triangulations are related by an integral piecewise-linear transformation, and the corresponding potential functions are related by its geometric lift in the sense of Berenstein and Zelevinsky.
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