Let $C$ be a projective curve (not necessarily reduced or irreducible). Let $f:C \hookrightarrow \mathbb{P}^n$ be a closed immersion. Suppose that the degree of $f(C)$ is equal to $e$. How many ways are there to embed $C$ in $\mathbb{P}^{n+1}$ such that the degree of the image is equal to $e$? Can we "parametrize" such embeddings in any way?

  • $\begingroup$ Many ways! Let us say that $C$ is smooth of genus $g$, and take, say, $e>2g+5$. You can find an embedding in $\mathbb{P}^3$ of degree $e$ -- but every line bundle of degree $e$ will give you plenty of embedding of degree $e$ in $\mathbb{P}^4$. $\endgroup$ – abx Jan 10 '14 at 15:54

These embeddings can indeed be parameterized. Explaining parameterizations of projective embeddings of a fixed degree is one of the main themes of the book: Geometry of Algebraic Curves: Volume I by Arbarello, Cornalba, Griffiths and Harris (ACGH).

You should especially read chapter IV: The Varieties of Special Linear Series on a Curve, where they describe the parameterization. Although the exposition in this chapter is for smooth curves over $\mathbb C$ the main approach generalizes. All that one needs is that the $Pic$ functor is representable.

Some results about "how many ways", i.e. the dimension of the parameterization can be found in chapter $V$. I don't know how these results generalize to non reduced or reducible curves.

The main approach of these kind of questions is as follows. If $g : C \to \mathbb P^n$ is a morphism then the pullback of the twisting sheaf $\mathcal O(1)$ on $\mathbb P^n$ gives a line bundle of degree equal to $e$. And if we write $\mathbb P^n = {\rm Proj \,} k[x_0,\ldots,x_n]$ then the $f^*(x_0),\ldots, f^*(x_n)$ are global section which will generate $f^*(\mathcal O(1))$. In fact from the linebundle $f^*(O(1))$ and the $n+1$ sections that generate it one can also contsruct the morphism $g$ again. So you see that your question is basically comes down to parameterizing linebundles together with a set of $n+1$ sections that generate it.

In particular the parameter space you are looking for will be an open subscheme of the scheme $G_e^n(C)$ in chapter IV of ACGH, it will not be all of $G_e^n(C)$ because you have to remove the closed part where the sections of de $g_e^n$ don't generate it.


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