4_6x

IV.6: Classification of Curves in \({\mathbf{P}}^3\)

IV.6.1. #to_work

A rational curve of degree 4 in \(\mathbf{P}^3\) is contained in a unique quadric surface \(Q\), and \(Q\) is necessarily nonsingular.

IV.6.2. #to_work

A rational curve of degree 5 in \(\mathbf{P}^3\) is always contained in a cubic surface, but there are such curves which are not contained in any quadric surface.

A curve of degree 5 and genus 2 in \(\mathbf{P}^3\) is contained in a unique quadric surface \(Q\). Show that for any abstract curve \(X\) of genus 2 , there exist embeddings of degree 5 in \(\mathbf{P}^3\) for which \(Q\) is nonsingular, and there exist other embeddings of degree 5 for which \(Q\) is singular.

IV.6.4. #to_work

There is no curve of degree 9 and genus 11 in \(\mathbf{P}^3\). ¹

IV.6.5. #to_work

If \(X\) is a complete intersection of surfaces of degrees \(a, b\) in \(\mathbf{P}^3\), then \(X\) does not lie on any surface of degree \(<\min (a, b)\).

IV.6.6. #to_work

Let \(X\) be a projectively normal curve in \(\mathbf{P}^3\), not contained in any plane. If \(d=6\), then \(g=3\) or 4 . If \(d=7\), then \(g=5\) or 6 . Cf. (II, Ex. 8.4) and (III, Ex. 5.6).

IV.6.7. #to_work

The line, the conic, the twisted cubic curve and the elliptic quartic curve in \(\mathbf{P}^3\) have no multisecants. Every other curve in \(\mathbf{P}^3\) has infinitely many multisecants. ²

IV.6.8. #to_work

A curve \(X\) of genus \(g\) has a nonspecial divisor \(D\) of degree \(d\) such that \(|D|\) has no base points if and only if \(d \geqslant g+1\).

IV.6.9. #to_work

* Let \(X\) be an irreducible nonsingular curve in \(\mathbf{P}^3\). Then for each \(m \gg>0\), there is a nonsingular surface \(F\) of degree \(m\) containing \(X\). ³

Footnotes

Hint: Show that it would have to lie on a quadric surface, then use (6.4.1).

Hint: Consider a projection from a point of the curve to \(\mathbf{P}^2\).

Hint: Let \(\pi: \tilde{\mathbf{P}} \rightarrow \mathbf{P}^3\) be the blowing-up of \(X\) and let \(Y=\pi^{-1}(X)\). Apply Bertini’s theorem to the projective embedding of \(\tilde{\mathbf{P}}\) corresponding to \(\mathcal{I}_Y \otimes \pi^* \mathcal{O}_{\mathbf{P}^3}(m)\).