I.2.12

This has H = 1.

HAPPY
For the other direction, let $$ Q = (Q_0,..., Q_N) \in V(a)$$, the task is to fine $$(a_0, ..., a_n)$$ such that $$Q = (M_0(a_i),...,M_N(a_i))$$; use induction on $$n$$; the base case $$n =1 $$ require that k is algebraically closed, so you can take dth roots.
 * for part a) take $$ f \in a$$ and break it into homogeneous parts $$f = \sum_i f_i$$ and argue (using $$ f(M_0, ..., M_N) = 0$$) that in fact each $$ f_i \in a$$; mod out by $$ a$$ and show you get an inclusion into a domain.
 * for part b), the easy direction is $$ im \rho_d \subset V(a)$$.
 * Let $$m_j$$ ($$ j = 0,1, ..., n$$) be the index such that $$ M_{m_j} = x_j^d$$; argue for some j, $$ Q_{m_j} \ne 0$$.
 * if some $$Q_{m_j}$$ is zero then can set $$ a_j = 0$$ i.e. can reduce to n-1 variables and apply induction hypothesis.
 * thus reduce to case where all $$Q_{m_j}$$ are all nonzero; for fixed j get a set of equations (indexed by i)

M_i(x_0/x_j, ...., x_n/x_j) = Q_i <\center>


 * there are consistent! since ($$ Q \in Z(a)$$).
 * argue all $$ Q_i$$ are nonzero (not just the $$Q_{m_j}$$)
 * then argue that by continuing to divide by certain equations can reduce the number of variables and solve (similar to how base case is proved).


 * for part c), show for a general closed set $$Z(b) \subset \mathbb{P}^n$$ that $$\rho_d(Z(b)) = Z(\theta^{-1}(b)) \cap Z(a)$$ hence $$\rho_d$$ is a closed map and consequently a homeomorhpism (i.e. had bijective and continuous, now showed the inverse was continuous).


 * part d) is straightforward.