@antonia To be fair, there is so much variety in that book at least some of hit has to be dangerous, right?
Fun fact: When I was in grad school my algebraic geometry teacher assigned us a homework question that required us to compute the "blow-up" of a "pentagon." He warned us not to Google it to avoid ending up on FBI watch-lists.
#Macaulay2 is a #ComputerAlgebra System devoted to supporting research in #AlgebraicGeometry and commutative algebra, whose creation has been funded by the National Science Foundation since 1992.
Macaulay2 includes core algorithms for computing Gröbner bases and graded or multi-graded free resolutions of modules over quotient rings of graded or multi-graded polynomial rings with a monomial ordering. The core algorithms are accessible through a versatile high level interpreted user language with a powerful debugger supporting the creation of new classes of mathematical objects and the installation of methods for computing specifically with them. Macaulay2 can compute Betti numbers, Ext, cohomology of coherent sheaves on projective varieties, primary decomposition of ideals, integral closure of rings, and more.
An affine variety can be embedded into proj space in many ways. Do they all give the same Betti tables? Or are the Betti tables somehow related? So much I've found starts w projective varieties / graded rings. Are there good keywords to search for to understand 👆?