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Deformation theory from the point of view of fibered categories
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We give an exposition of the formal aspects of deformation theory in the language of fibered categories, instead of the more traditional one of functors. The main concepts are that of tangent space to a deformation problem, obstruction theory, versal and universal formal deformations. We include proofs of two key results: a versione of Schlessinger's Theorem in this context, and the Ran--Kawamata vanishing theorem for obstructions. We accompany this with a detailed analysis of three important cases: smooth varieties, local complete intersection subschemes and coherent sheaves.
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Explicit bounds on common projective torsion points of elliptic curves
For elliptic curves with standard double covers and suitable reduction at a prime p, the number of common projective torsion points is at most 2p^3+8, with refinements and a conditional bad-reduction analogue.
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