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Variable stars in Local Group Galaxies - II. Sculptor dSph

We present the identification of 634 variable stars in the Milky Way dSph satellite Sculptor based on archival ground-based optical observations spanning $\sim$24 years and covering $\sim$ 2.5 deg$^2$. We employed the same methodologies as the "Homogeneous Photometry" series published by Stetson. In particular, we have identified and characterized one of the largest (536) RR Lyrae samples so far in a Milky Way dSph satellite. We have also detected four Anomalous Cepheids, 23 SX Phoenicis stars, five eclipsing binaries, three field variable stars, three peculiar variable stars located above the horizontal branch - near to the locus of BL Herculis - that we are unable to classify properly. Additionally we identify 37 Long Period Variables plus 23 probable variable stars, for which the current data do not allow us to determine the period. We report positions and finding charts for all the variable stars, and basic properties (period, amplitude, mean magnitude) and light curves for 574 of them. We discuss the properties of the RR Lyrae stars in the Bailey diagram, which supports the coexistence of subpopulations with different chemical compositions. We estimate the mean mass of Anomalous Cepheids ($\sim$1.5M$_{\odot}$) and SX Phoenicis stars ($\sim$1M$_{\odot}$). We discuss in detail the nature of the former. The connections between the properties of the different families of variable stars are discussed in the context of the star formation history of the Sculptor dSph galaxy.