MEGA is the acronym for Effective Methods in Algebraic Geometry (and its equivalent in Italian, French, Spanish, German, Russian, etc.), a series of roughly biennial conferences on computational and application aspects of Algebraic Geometry and related topics with very high standards. Previous meetings were held in 1990 (Castiglioncello, Italy), 1992 (Nice, France), 1994 (Santander, Spain), 1996 (Eindhoven, Nederlands), 1998 (St. Malo, France) 2000 (Bath, United Kingdom), 2003 (Kaiserslautern, Germany), 2005 (Porto Conte, Italy) and 2007 (Strobl, Austria).
Proceedings containing a selection of the papers and invited talks presented at previous Mega conferences have been published by Birkhäuser in the series Progress in Mathematics (volumes no. 94, 109 and 143), by the Journal of Pure and Applied Algebra (volumes no. 117 and 118, 139 and 164) and by the Journal of Symbolic Computation (volumes no. 39 3-4 and 42 1-2).
Refs for computing de Rham cohomology of a complex variety: http://arxiv.org/pdf/0905.2212v1
Stillman et al ed: Software for algebraic geometry
Title: Effective computation of Picard groups and Brauer-Manin obstruction of degree two K3 surfaces over number fields Authors: Brendan Hassett, Andrew Kresch, Yuri Tschinkel http://front.math.ucdavis.edu/1203.2214 Categories: math.AG Algebraic Geometry Comments: 15 pages MSC: 14G25 (primary), 14F22, 14J28 (secondary) Abstract: Using the Kuga-Satake correspondence we provide an effective algorithm for the computation of the Picard and Brauer groups of K3 surfaces of degree 2 over number fields
arXiv:1205.5896 Approximate computations with modular curves from arXiv Front: math.AG by Jean-Marc Couveignes, Bas Edixhoven This article gives an introduction for mathematicians interested in numerical computations in algebraic geometry and number theory to some recent progress in algorithmic number theory, emphasising the key role of approximate computations with modular curves and their Jacobians. These approximations are done in polynomial time in the dimension and the required number of significant digits. We explain the main ideas of how the approximations are done, illustrating them with examples, and we sketch some applications in number theory.
nLab page on Computational algebraic geometry