Is every projective variety the zero set of a system equations which are homogeneous polynomials of the same degree?

Question

Is every projective variety the zero set of a system equations which are homogeneous polynomials of the same degree?

Answer 1

Yes, you can assume that all defining equations have the same degree! Here is the trick:
Suppose that the variety $V\subset \mathbb P^n_k$ is defined by $s$ homogeneous polynomials $f_i(x_0,\cdots,x_n)\in k[x_0,\cdots,x_n]$ of degrees $\deg f_i=d_i$, namely $$V=Z(f_i\vert i=1,\cdots,s)$$ Now let $d=\max d_i$. The trick is then to realize that each equation $f_i=0$ is equivalent to the system of $n+1$ equations $$f_i\cdot x_j^{d-d_i}=0 \;(j=0,\cdots,n)$$ We thus see that $V$ is defined by the system of $s(n+1)$ equations $f_i\cdot x_j^{d-d_i}=0$ all of degree d: $$V=Z(f_i\cdot x_j^{d-d_i}\vert i=1,\cdots17\;;j=0,\cdots,n)$$ (And note that this trick works whether "irreducible" is part of your definition of variety or not.)