## IV.3: Embeddings in Projective Space $\star$

:::{.remark}
A summary of major results:

- For $D\in \Div(C)$ with $g = g(C)$, 
  - $D$ is ample iff $\deg D > 0$.
  - $D$ is BPF iff $\deg D\geq 2g$.
  - $D$ is very ample iff $\deg D \geq 2g+1$.
- Being very ample is equivalent to being a hyperplane section under a projective embedding.
- Divisors $D\in \Div(\PP^n)$ are ample iff very ample iff $\deg D \geq 1$.
  - E.g. if $E$ is elliptic then $D$ is very ample if $\deg D \geq 3$, and for hyperelliptic, very ample if $\deg D\geq 5$.
- If $D$ is very ample then $\deg \phi(X) = \deg D$.
- Curves $C \subseteq \PP^n$ for $n\geq 4$ can be projected away from a point $p\not \in X$ to get a closed immersion into $\PP^m$ for some $m\leq n-1$.
  So any curve is birational to a nodal curve in $\PP^2$.
- Genus of normalizations of nodal curves: $g = {1\over 2}(d-1)(d-2)-\size\ts{\text{nodes}}$.
- Any curve embeds into $\PP^3$, and maps into $\PP^2$ with at worst nodal singularities.
:::

:::{.remark}
Main result: any curve can be embedded in $\PP^3$, and is birational to a nodal curve in $\PP^2$.
Some recollections:

- **Very ample line bundles**: $\mcl \in \Pic(X)$ is very ample if $\mcl \cong \OO_X(1)$ for some immersion of $f: X\embeds \PP^N$.
- **Ample**: $\mcl$ is ample when $\forall \mcf\in \Coh(X)$, the twist $\mcf \tensor \mcl^n$ is globally generated for $n \gg 0$.
- **(Very) ample divisors**: $D\in \Div(X)$ is (very) ample iff $\mcl(D)\in \Pic(X)$ is (very) ample.
- **Linear systems**: a linear system is any set $S \leq \abs{D}$ of effective divisors yielding a linear subspace.
- **Base points**: $P$ is a base point of $S$ iff $P \in \supp D$ for all $D\in S$.
- **Secant lines**: the secant line of $P, Q\in X$ is the line in $\PP^N$ joining them.
- **Tangent lines**: at $P\in X$, the unique line $L \subseteq \PP^N$ passing through $p$ such that $\T_P(L) = \T_P(X) \subseteq \T_P(\PP^N)$.
- **Nodes**: a singularity of multiplicity 2.
  - $y^2 = x^3 + x^2$ is a **node**.
  - $y^2 = x^3$ is a **cusp**.
  - $y^2 = x^4$ is a **tacnode**.
- **Multisecant**: for $X \subseteq \PP^3$, a line meeting $X$ in 3 or more distinct points.
- A **secant with coplanar tangent lines** is a secant through $P, Q$ whose tangent lines $L_P, L_Q$ lie in a common plane, or equivalently $L_P$ intersects $L_Q$.

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:::{.exercise title="II.8.20.2"}
Show that by Bertini's theorem there are irreducible smooth curves of degree $d$ in $\PP^2$ for any $d$.
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:::{.exercise title="?"}
\envlist
Show that

- $\mcl$ is ample iff $\mcl^n$ is very ample for $b \gg 0$.
- $\abs D$ is basepoint free iff $\mcl(D)$ is globally generated.
- If $D$ is very ample, then $\abs{D}$ is basepoint free.
- If $D$ is ample, $nD \sim H$ a hyperplane section for a projective embedding for some $n$.
- If $g(X) = 0$ then $D$ is ample iff very ample iff $\deg D > 0$.
- If $D$ is very ample and corresponds to a closed immersion $\phi: X\injects \PP^n$ then $\deg \phi(X) = \deg D$.
- If $XS$ is elliptic, any $D$ with $\deg D = 3$ is very ample and $\dim \abs{D} = 2$, and so can be embedded into $\PP^2$ as a cubic curve.
- Show that if $g(X) = 1$ then $D$ is very ample iff $\deg D \geq 3$.
- Show that if $g(X) = 2$ and $\deg D = 5$ then $D$ is very ample, so any genus 2 curve embeds in $\PP^3$ as a curve of degree 5.

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:::{.exercise title="Prop 3.1: when a linear system yields a closed immersion into $\PP^N$"}
Let $D\in \Div(X)$ for $X$ a curve and show

- $\abs{D}$ is basepoint free iff $\dim\abs{D-P} = \dim\abs{D} - 1$ for all points $p\in X$.
- $D$ is very ample iff $\dim\abs{D-P-Q} = \dim\abs{D} - 2$ for all points $P, Q\in X$.

> Hint: use the SES $\mcl(D-P)\injects \mcl(D) \surjects k(P)$ where $k(P)$ is the skyscraper sheaf at $P$.

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:::{.exercise title="Cor 3.2"}
Let $D\in \Div(X)$.

- If $\deg D \geq 2g(X)$ then $\abs{D}$ is basepoint free.
- If $\deg D \geq 2g(X) + 1$ then $D$ is very ample.
- $D$ is ample iff $\deg D > 0$
- This bounds is not sharp.

> Hint: apply RR. For the bound, consider a plane curve $X$ of degree 4 and $D = X.H$.

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:::{.remark}
Idea behind embedding in $\PP^3$: embed into $\PP^n$ and project away from a point in the complement.
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:::{.exercise title="3.4, 3.5, 3.6"}
Let $X \subseteq \PP^N$ be a curve and $O\not\in X$, let $\phi:X\to \PP^{n-1}$ be projection away from $O$.
Then $\phi$ is a closed immersion iff

- $O$ is not on any secant line of $X$, and
- $O$ is not on any tangent line of $X$.

Show that if $N\geq 4$ then there exists such a point $O$ yielding a closed immersion into $\PP^{N-1}$.
Conclude that any curve can be embedded into $\PP^3$.

> Hint: $\dim\mathrm{Sec}(X) \leq 3$ and $\dim \mathrm{Tan}(X) \leq 2$.

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:::{.proposition title="3.7"}
Let $X \subseteq \PP^3$, $O\not\in X$, and $\phi: X\to \PP^2$ be the projection from $O$.
Then $X\birational \phi(X)$ iff $\phi(X)$ is nodal iff the following hold:

- $O$ is only on finitely many secants of $X$,
- $O$ is on no tangents,
- $O$ is on no multisecant,
- $O$ is on no secant with coplanar tangent lines.

:::

> Skipped things around Prop 3.8.
> The hard part: showing not every secant is a multisecant, and not every secant has coplanar tangent lines.
> Skipped strange curves.

:::{.remark}
Classifying all curves: any curve is birational to a nodal plane curve, so study the family $\mcf_{d, r}$ of plane curves of degree $d$ and $r$ nodes.
The family $\mcf_d$ of all plane curves is a linear system of dimension
\[
\dim \abs{\mcf_d} = {d(d+3)\over 2}
.\]
For any such curve $X$, consider its normalization $\nu(X)$, then
\[
g(\nu(X)) = {(d-1)(d-2)\over 2} - r
.\]
Thus for $\mcf_{d, r}$ to be nonempty, one needs
\[
0 \leq r \leq {(d-1)(d-2) \over 2}
.\]
Both extremes can occur: $r=0$ follows from Bertini, and $r = {(d-1)(d-2)\over 2}$ by embedding $\PP^1\injects \PP^d$ as a curve of degree $d$ and projecting down to a nodal curve in $\PP^2$ of genus zero.
Severi states and Harris proves that for every $r$ in this range $\mcf_{d, r}$ is irreducible, nonempty, and $\dim \mcf_{d, r} = {d(d+3)\over 2} - r$.
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