Irreducibility of Random Hilbert Schemes

We prove that a random Hilbert scheme that parametrizes the closed subschemes with a fixed Hilbert polynomial in some projective space is irreducible and nonsingular with probability greater than $0.5$. To consider the set of nonempty Hilbert schemes as a probability space, we transform this set into a disjoint union of infinite binary trees, reinterpreting Macaulay's classification of admissible Hilbert polynomials. Choosing discrete probability distributions with infinite support on the trees establishes our notion of random Hilbert schemes. To bound the probability that random Hilbert schemes are irreducible and nonsingular, we show that at least half of the vertices in the binary trees correspond to Hilbert schemes with unique Borel-fixed points.

The Most Complete Photometric Analysis of 548 CALIFA Galaxies

We present an extensive photometric catalog for 548 CALIFA galaxies observed as of the summer of 2015. CALIFA is currently lacking photometry matching the scale and diversity of its spectroscopy; this work is intended to meet all photometric needs for CALIFA galaxies while also identifying best photometric practices for upcoming integral field spectroscopy surveys such as SAMI and MaNGA. This catalog comprises gri surface brightness profiles derived from Sloan Digital Sky Survey (SDSS) imaging, a variety of non-parametric quantities extracted from these pro files, and parametric models fitted to the i-band pro files (1D) and original galaxy images (2D). To compliment our photometric analysis, we contrast the relative performance of our 1D and 2D modelling approaches. The ability of each measurement to characterize the global properties of galaxies is quantitatively assessed, in the context of constructing the tightest scaling relations. Where possible, we compare our photometry with existing photometrically or spectroscopically obtained measurements from the literature. Close agreement is found with Walcher et al. (2014), the current source of basic photometry and classifications of CALIFA galaxies, while comparisons with spectroscopically derived quantities reveals the effect of CALIFA's limited field of view compared to broadband imaging surveys such as the SDSS. The colour-magnitude diagram, star formation main sequence, and Tully-Fisher relation of CALIFA galaxies are studied, to give a small example of the investigations possible with this rich catalog. We conclude with a discussion of points of concern for ongoing integral field spectroscopy surveys and directions for future expansion and exploitation of this work.