TITLE: Independent conditions of linear system
QUESTION [1 upvotes]: I am reading the book on "geometry of algebraic curves" by Harris ect. And I met this statement here:
My question is what it means that the conditions are independent, i.e., what I need to show here? And usually, we have a (complete) linear system given by a divisor $|D|$, or a line bundle or an invertible sheaf on a curve. But why here is it sheaf on $\mathbb{P}^2$?

REPLY [1 votes]: Think about a curve in $\mathbb P^2$ and a line; they always intersect, so the set of curves meeting the line is the complete linear system. On the other hand, the space of curves passing through a given point has codimension $1$ in $|\mathcal O_{\mathbb P^2}(d-3)|$. So the space of curves passing through $\delta$ points is obtained by intersecting $\delta$ of these codim. $1$ subspaces. If the subspaces are general, one expects the intersection to have codimension $\delta$, but it may be that the intersection is larger; in other words, the set of curves passing through $p_1,...,p_k$ also all pass through $p_{k+1}$ (after reordering). So what you are trying to prove is that the nodes of $\Gamma$ are in sufficiently general position that the associated subspaces intersect generically.
When one says that a linear system of hypersurfaces in $\mathbb P^n$ (e.g. curves in $\mathbb P^2$) cuts out a certain linear series on a curve, we mean the restrictions of those hypersurfaces. Assuming that $\phi:C \to \mathbb P^2$ is the normalization, and assuming that $\Delta$ refers to the divisor $\phi^{*}(p_1 + \cdots + p_\delta)$, this is saying that although the canonical bundle on $C$ is (by adjunction) the restriction of $\mathcal O_{\mathbb P^2}(d-3)$, to get all of the sections one can just restrict the subspace of curves passing through all of the nodes (as opposed to all degree $(d-3)$ curves). A divisor in this linear system is obtained by choosing a curve in $\Sigma$, computing its intersection with $\phi(C)$, considering that intersection as a divisor on $C$, and throwing away the base locus $\Delta$.
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