What is $S_d$ in algebraic geometry?

Question

I'm trying to read algebraic geometry on my own by doing homeworks on course hold in 2003. One of the problem is the following:

Let $k$ be a field, $S=k[T_0,\ldots,T_r]$, $\mathbb{P}=\mathbb{P}_k^r=\operatorname{Proj}(S)$ and $\mathcal O$ the structure sheaf of $\mathbb{P}$. Let $r=3$ and let $f\in S_d,g\in S_e$ be two relatively prime homogeneous forms. Let $X=V_+(f)\cap V_+(g)$. Compute the characteristic $\chi (O_X)$.

But what does the symbols $S_d,S_e$ means and how to do the problem?

Answer 1

Now that the the meaning of the notations is clarified, here is a solution to the actual problem.
We have a short exact sequence of coherent sheaves on $P=\mathbb P^3_k$:

$$0\to\mathcal O_P (-d-e)\to \mathcal O_P (-d)\oplus\mathcal O_P (-e) \to \mathcal O_P\to\mathcal O_Z\to 0$$ The first non trivial map is $h\mapsto (hg,-hf)$ and the second is $(u,v)\mapsto uf+vg$
Exactness at $\mathcal O_P (-d)\oplus\mathcal O_P (-e)$ results from $f,g$ being relatively prime.

Taking Euler characteristic, which is additive in short exact sequences, we get: $$\chi(Z, \mathcal O_Z) =\chi(P,\mathcal O_P (-d-e))- \chi(P,\mathcal O_P (-d))-\chi(P,\mathcal O_P (-e))+\chi(P,\mathcal O_P )$$ Finally, remembering that $\chi(P,\mathcal O_P (k))=\binom {k+3}{3}$, we obtain the desired formula $$\chi(Z, \mathcal O_Z) = \binom {-d-e+3}{3}-\binom {-d+3}{3}-\binom {-e+3}{3}+1$$

Answer 2

$S_d \subseteq S$ is the space of homogeneous polynomials of degree $d,$ i.e. the $k$-span of all monomials $T_0^{a_0}\cdots T_r^{a_r}$ with $a_0,\ldots,a_r \in \mathbb N_0$ and $a_0 +\cdots+a_r =d.$