 Magma 
Software package designed for computations in algebra, number theory, algebraic geometry and algebraic combinatorics. 
 MacAulay2 
Software system devoted to supporting research in algebraic geometry and commutative algebra.

 Bertini  Software for solving polynomial systems. 
Packages  
Fermat.zip  The library contains several Magma functions computing the birational parametrizations of the Fermat cubic hypersurfaces (of even dimension) and their inverses as described in the paper "Rational points on even dimensional Fermat cubics". See also Fermat cubics on GitHub. 

P1xPn.zip  Magma library computing the nef cones and the Mori chamber decompositions of blowups of ℙ^{1} × ℙ^{n} in general points. See also Cox rings of blowups of multiprojective spaces on GitHub. 

Grass_Magma.zip  Magma function computing the Grassmannian of kplanes in an ndimensional projective space.  
Dual_Magma.zip  Magma function computing the dual of a projective hypersurface.  
Tensor_Decomposition.zip  Magma library implementing the algorithms for tensor and polynomial decomposition in "Decomposition algorithms for tensors and polynomials", by A. Laface, A. Massarenti and R. Rischter.  
Toric_Def.zip  Magma package checking whether a toric variety is defective as described in "On secant defectiveness and identifiability of SegreVeronese varieties", by A. Laface, A. Massarenti and R. Rischter.  
Comon1.0.m2  Macaulay 2 package checking whether a homogeneous polynomial satisfies Comon's conjecture as described in "On Comon's and Strassen's conjectures", by A. Casarotti, A. Massarenti and M. Mella. 

MCD_SBLD.zip  Magma package checking whether the Mori chamber and the stable base locus decomposition of a Mori dream space coincide as described in "On Mori chamber and stable base locus decompositions", by A. Laface, A. Massarenti and R. Rischter.  
Movable_Complete_Forms.zip  Maple scripts computing the extremal rays of the movable cones of spaces of complete forms as described in "On the birational geometry of spaces of complete forms I: collineations and quadrics" and "On the birational geometry of spaces of complete forms II: skewforms".  
Identifiability.m2  Macaulay 2 package implementing the identifiability criteria in "Effective identifiability criteria for tensors and polynomials" by A. Massarenti, M. Mella and G. Staglianò.  
Movable.mw  Maple script computing the extremal rays of the movable cone of a Mori dream space from the generators of its Cox ring, by A. Massarenti and R. Rischter. 