IV.3: Embeddings in Projective Space
Main result: any curve can be embedded in \({\mathbf{P}}^3\), and is birational to a nodal curve in \({\mathbf{P}}^2\). Some recollections:
- Very ample line bundles: \({\mathcal{L}}\in \operatorname{Pic}(X)\) is very emply if \({\mathcal{L}}\cong {\mathcal{O}}_X(1)\) for some immersion of \(X\) into some \({\mathbf{P}}^N\).
- Ample: \({\mathcal{L}}\) is ample when \(\forall {\mathcal{F}}\in {\mathsf{Coh}}(X)\), the twist \({\mathcal{F}}\otimes{\mathcal{L}}^n\) is globally generated for \(n \gg 0\).
- (Very) ample divisors: \(D\in \operatorname{Div}(X)\) is (very) ample iff \({\mathcal{L}}(D)\) is (very) ample.
- Linear systems: a linear system is any set \(S \leq {\left\lvert {D} \right\rvert}\) of effective divisors yielding a linear subspace.
- Base points: \(P\) is a base point of \(S\) iff \(P \in \mathop{\mathrm{supp}}D\) for all \(D\in S\).
- Secant lines: the secant line of \(P, Q\in X\) is the line in \({\mathbf{P}}^N\) joining them.
- Tangent lines: at \(P\in X\), the unique line \(L \subseteq {\mathbf{P}}^N\) passing through \(p\) such that \({\mathbf{T}}_P(L) = {\mathbf{T}}_P(X) \subseteq {\mathbf{T}}_P({\mathbf{P}}^N)\).
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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 \(X \subseteq {\mathbf{P}}^3\), 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 \(L_Q\).
Show that by Bertini’s theorem there are irreducible smooth curves of degree \(d\) in \({\mathbf{P}}^2\) for any \(d\).
\envlist
Show that
- \({\mathcal{L}}\) is ample iff \({\mathcal{L}}^n\) is very ample for \(b \gg 0\).
- \({\left\lvert {D} \right\rvert}\) is basepoint free iff \({\mathcal{L}}(D)\) is globally generated.
- If \(D\) is very ample, then \({\left\lvert {D} \right\rvert}\) 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 \(\phi: X\hookrightarrow{\mathbf{P}}^n\) then \(\deg \phi(X) = \deg D\).
- If \(XS\) is elliptic, any \(D\) with \(\deg D = 3\) is very ample and \(\dim {\left\lvert {D} \right\rvert} = 2\), and so can be embedded into \({\mathbf{P}}^2\) 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 \({\mathbf{P}}^3\) as a curve of degree 5.
Let \(D\in \operatorname{Div}(X)\) for \(X\) a curve and show
- \({\left\lvert {D} \right\rvert}\) is basepoint free iff \(\dim{\left\lvert {D-P} \right\rvert} = \dim{\left\lvert {D} \right\rvert} - 1\) for all points \(p\in X\).
- \(D\) is very ample iff \(\dim{\left\lvert {D-P-Q} \right\rvert} = \dim{\left\lvert {D} \right\rvert} - 2\) for all points \(P, Q\in X\).
Hint: use the SES \({\mathcal{L}}(D-P)\hookrightarrow{\mathcal{L}}(D) \twoheadrightarrow k(P)\) where \(k(P)\) is the skyscraper sheaf at \(P\).
Let \(D\in \operatorname{Div}(X)\).
- If \(\deg D \geq 2g(X)\) then \({\left\lvert {D} \right\rvert}\) 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 \(D = X.H\).
Idea behind embedding in \({\mathbf{P}}^3\): embed into \({\mathbf{P}}^n\) and project away from a point in the complement.
Let \(X \subseteq {\mathbf{P}}^N\) be a curve and \(O\not\in X\), let \(\phi:X\to {\mathbf{P}}^{n-1}\) 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 \({\mathbf{P}}^{N-1}\). Conclude that any curve can be embedded into \({\mathbf{P}}^3\).
Hint: \(\dim\mathrm{Sec}(X) \leq 3\) and \(\dim \mathrm{Tan}(X) \leq 2\).
Let \(X \subseteq {\mathbf{P}}^3\), \(O\not\in X\), and \(\phi: X\to {\mathbf{P}}^2\) be the projection from \(O\). Then \(X\overset{\sim}{\dashrightarrow}\phi(X)\) 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.
Classifying all curves: any curve is birational to a nodal plane curve, so study the family \({\mathcal{F}}_{d, r}\) of plane curves of degree \(d\) and \(r\) nodes. The family \({\mathcal{F}}_d\) of all plane curves is a linear system of dimension \begin{align*} \dim {\left\lvert {{\mathcal{F}}_d} \right\rvert} = {d(d+3)\over 2} .\end{align*} For any such curve \(X\), consider its normalization \(\nu(X)\), then \begin{align*} g(\nu(X)) = {(d-1)(d-2)\over 2} - r .\end{align*} Thus for \({\mathcal{F}}_{d, r}\) to be nonempty, one needs \begin{align*} 0 \leq r \leq {(d-1)(d-2) \over 2} .\end{align*} Both extremes can occur: \(r=0\) follows from Bertini, and \(r = {(d-1)(d-2)\over 2}\) by embedding \({\mathbf{P}}^1\hookrightarrow{\mathbf{P}}^d\) as a curve of degree \(d\) and projecting down to a nodal curve in \({\mathbf{P}}^2\) of genus zero. Severi states and Harris proves that for every \(r\) in this range \({\mathcal{F}}_{d, r}\) is irreducible, nonempty, and \(\dim {\mathcal{F}}_{d, r} = {d(d+3)\over 2} - r\).
Stopped at exercises.