Algebraic Geometry
Magma covers affine and projective varieties, schemes, coherent sheaves, and divisors over arbitrary fields. Tools include Riemann-Roch, intersection theory, blow-ups, and toric geometry.
Contributors
A major package for algebraic surfaces providing formal desingularization, the calculation of adjoints, and rational parameterization was developed by Tobias Beck (RICAM, Linz). He also implemented a package for computing with algebraic power series. This work was done while he was a student of Josef Schicho.
A package for working with divisors on varieties has been developed by Martin Bright (American University of Beirut), Gavin Brown (Loughborough), Mike Harrison (Magma) and Andrew Wilson (Edinburgh). The functionality includes decomposition into irreducible components, Riemann–Roch spaces, canonical divisors and (surface) intersection numbers.
The machinery for working with Hilbert series of polarised varieties and the associated databases of K3 surfaces and Fano 3-folds has been constructed by Gavin Brown (Warwick).
A package for toric geometry together with a package for polytopes and polyhedra were developed by Jarosław Buczyński (Texas A&M), Gavin Brown (Loughborough) and Alexander Kasprzyk (Magma and Imperial College).
Functions for computing Shioda invariants for genus 3 hyperelliptic curves, reconstructing models for a curve from such invariants and computing geometric automorphism groups have been contributed by Reynald Lercier (DGA, Rennes) and Christophe Ritzenthaler (Luminy).
Christian Neurohr (Universität Oldenburg) has contributed his package for computing with Riemann surfaces.
Jana Pilnikova (Univerzita Komenskeho, Bratislava) (while a student of Josef Schicho in Linz) contributed code for the parameterization of degree 8 and 9 Del Pezzo surfaces, jointly written with Willem de Graaf (Trento).
Miles Reid (Warwick) has been heavily involved in the design and development of a database of K3 surfaces within Magma.
Carlos Rito (Universidade de Tras-os-Montes e Alto Douro, Portugal) has contributed code for computing with linear systems (2023). This includes the linear system of an affine plane curve which has non-ordinary singularities.
Josef Schicho (RICAM, Linz) has played a major role in the design and implementation of the algebraic surfaces package. In particular, Josef has also implemented several of the modules for rational surface parameterization.
A function that finds the intersection multiplicities for all intersection points of two plane curves was adapted into Magma from code provided by Chris Smyth (Edinburgh).
Andrew Wilson (Edinburgh) has contributed a package to compute the log canonical threshold for singular points on a curve.