Some syzygies of the generators of the ideal of a border basis scheme

A border basis scheme is an affine scheme that can be viewed as an open subscheme of the Hilbert scheme of \mu points of affine n-space. We study syzygies of the generators of a border basis scheme's defining ideal. These generators arise as the entries of the commutators of certain matrices (the "generic multiplication matrices"). We consider two families of syzygies that are closely connected to these matrices: The first arises from the Jacobi identity, and the second from the fact that the trace of a commutator is 0. Several examples of both types of syzygy are presented, including a proof that the border basis schemes in case n = 2 are complete intersections.Comment: Revised in accordance with referee comments: exposition improved, typos & minor errors fixed. To appear in Collectanea Mathematica. 25 page

Color and Variability Characteristics of Point Sources in the Faint Sky Variability Survey

We present an analysis of the color and variability characteristics for point sources in the Faint Sky Variability Survey (FSVS). The FSVS cataloged ~23 square degrees in BVI filters from ~16--24 mag to investigate variability in faint sources at moderate to high Galactic latitudes. Point source completeness is found to be >83% for a selected representative sample (V=17.5--22.0 mag, B-V=0.0--1.5) containing both photometric B, V detections and 80% of the time-sampled V data available compared to a basic internal source completeness of 99%. Multi-epoch (10--30) observations in V spanning minutes to years modeled by light curve simulations reveal amplitude sensitivities to 0.015--0.075 mag over a representative V=18--22 mag range. Periodicity determinations appear viable to time-scales of an order 1 day or less using the most sampled fields (~30 epochs). The fraction of point sources is found to be generally variable at 5--8% over V=17.5--22.0 mag. For V brighter than 19 mag, the variable population is dominated by low amplitude (<0.05 mag) and blue (B-V<0.35) sources, possibly representing a population of gamma Doradus stars. Overall, the dominant population of variable sources are bluer than B-V=0.65 and have Main Sequence colors, likely reflecting larger populations of RR Lyrae, SX Phe, gamma Doradus, and W UMa variables.Comment: 34 pages, 16 figures, accepted in A