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Assume a projective scheme $X_{k_1,\dots,k_r}\subset\mathbb{P}^n$ is given as the set of common solutions of homogeneous polynomials $F_1(x_0,\dots,x_n),\dots,F_s(x_0,\dots,x_n)$, where the $F_i$ depends on parameters $k_1,\dots,k_r$ varying in the base field.

Does there exist a computer algebra software that can compute the dimension and the degree of $X_{k_1,\dots,k_r}$ as functions of $k_1,\dots,k_r$?

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In addition to Magma (tagged in your question), Singular (cited in Jürgen Böhm's comment), there's also Macaulay2. AFAIK, all provide methods for calculating the required Gröbner basis.

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  • $\begingroup$ Could you give a reference for Macaulay2 of a documentation page where computing gröbner bases with parameters is described? Note that this is an extension of the usual concept of gröbner base: If you have $R=\mathbb{Q}[a][x]$ as basering with parameter $a$ and a polynom $f = a x^2 + x + 1$ then a gröbner base with parameters would give a list $[a \neq 0, f]$ as a first base and $[a=0, x+1]$ as a second base. This is useful in many applications, but I have not found a command in Macaulay2 to do this. The same applies to your Magma link. $\endgroup$ Jan 13 '21 at 17:18

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