I'm following a course in algebraic geometry and have a question about the fact that every non-singular curve can be embedded in $\mathbb{P}^3$. You should be aware that I have almost no background in algebraic geometry. When I assume that the curve $X$ lies already in some $\mathbb{P}^m$ it is doable. It is done by taking a point $P \in \mathbb{P}^m \backslash X$ outside the curve and project the curve from the point onto a hyperplane $H \cong \mathbb{P}^{m-1}$, this works when $m \geq 4$. The conditions $P$ has to fulfil are the following:
- $P$ is not on any secant line of $X$,
- $P$ is not on any tangent line of $X$.
By my very limited background it is hard to understand many things from the book of Hartshorne. Does anyone has a good reference on this subject? I would also like to ask what divisors and linear systems precisely do and how they come into play?
Note: This is a follow-up to a previous post on the same subject. The new question is specifically about the role of divisors in this construction. This issue was raised, but not addressed, in the comments beneath an answer in the previous post, which is why it is asked as a separate question here.
An affine curve is $f(x, y) = 0$ in $\mathbb{C}^2$. Its normalization is nonsingular by basic commutative algebra. Find your own reference.
Any projective curve $C$ in $\mathbb{P}^n$ is the union of standard affine pieces. You can take the normalization of each of these affine pieces separately, say$$\widetilde{C}_i \to C_i.$$Given the affine normalization $\widetilde{C}_i$, you can put it in $\mathbb{P}^{N_i}$ in any stupid way. Then the fiber product of all these $\widetilde{C}_i$ is contained in product $\mathbb{P}^{N_i}$ Segre product in $\mathbb{P}^M$—much bigger$$M = \prod (N_i + 1) - 1$$or something. The birational component is then nonsingular, because it has affine pieces dominating each of the $\widetilde{C}_i$. This proof is given somewhere in Shafarevich's book on algebraic geometry.