## IV.6: Classification of Curves in $\PP^3$

### 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.

### IV.6.3.  #to-work

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 $\mathbf{P}^3$.[^hint.4.6.4]

[^hint.4.6.4]: 
Hint: Show that it would have to lie on a quadric surface, then use (6.4.1).


### 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.[^4.6.7.hint]

[^4.6.7.hint]: 
Hint: Consider a projection from a point of the curve to $\mathbf{P}^2$.


### 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 $X$.[^hint.4.6.9]

[^hint.4.6.9]: 
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 $\mathcal{I}_Y \otimes \pi^* \mathcal{O}_{\mathbf{P}^3}(m)$.