What is the multidegree of a curve $C \hookrightarrow \mathbb{P}^n \times \mathbb{P}^m$?
I'm reading Notes on stable maps and quantum cohomology by W. Fulton and R. Pandharipande, and on page 14, there is the sentence
Let $H$ be the Hilbert scheme of genus $g$ curves in $\mathbb{P}(W) \times \mathbb{P}^r$ of multidegree $(e,d)$.
I know what the Hilbert scheme is, but afaik one needs to specify a Hilbert polynomial $P$. Fixing the (arithmetic) genus defines the constant term, because $g = (-1)^d(P(0) - 1)$, where $d$ is the dimension. The degree of $P$ is given by saying that we consider curves, i.e. $\deg(P) = 1$. So the leading coefficient is still missing.
Of course to define the Hilbet scheme (or even the Hilbert polynomial $P$) we also need to specify a very ample sheaf, or a closed embedding $\mathbb{P}^n \times \mathbb{P}^m \to \mathbb{P}^N$ for some $N > 0$. I actually have two guesses here, but I don't know which is it:
- simply take the Segre embedding. Here I have no clue what the multidegree could mean.
- take the embedding defined by the very ample invertible sheaf $\mathcal{O}_{\mathbb{P}^n}(e) \boxtimes \mathcal{O}_{\mathbb{P}^m}(d)$, and maybe fix $1$ as the leading coefficient for $P$?