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

:::{.remark}
- Smooth curves of genus $g$ and degree $d$ in $\PP^3$, $M_g^d$, are parameterized by a finite union of quasiprojective varieties.
- $\dim M_g^d$ is unsolved, it's not even clear for which $d,g$ this is empty.
- If $g\geq 2$ then $X$ has a nonspecial very ample divisor of degree $d$ iff $d\geq g+2$.
  - Proof: such a $D$ satisfies $h^0(D) = d-g$ and induces $X\embeds \PP^{d-g}$.
  Since $X\not\cong \PP^1$ this forces $d-g > 1$, so $d\geq g+2$ with equality iff $X$ is a plane curve of degree $d$.
  Then $\omega_X = \OO_X(d-3)$ and $D$ is nonspecial so $d \leq 3$, forcing $g=0,1$. $\contradiction$
- There exists a curve $X$ of degree $d$ and genus $g$ in $\PP^3$ with nonspecial hyperplane section $D$ iff  
  - $g=0, d\geq 1$,
  - $g=1, d\geq 3$,
  - $g\geq 2, d\geq g+3$.
- If $X \subseteq \PP^3$ is a curve not lying in a plane where the hyperplane section $D$ is special, then $d\geq 6$ and $g\geq {1\over 2}d+1$.
  The only such curve with $d=6$ is the canonical curve of genus $4$.
- Castelnuovo: if $X \subset \PP^3$ is a curve of degree $d$, genus $g$, not contained in a plane, then $d\geq 3$ and
\[
g\leq \begin{cases}\frac{1}{4} d^2-d+1 & \text { if } d \text { is even } \\ \frac{1}{4}\left(d^2-1\right)-d+1 & \text { if } d \text { is odd. }\end{cases}
,\]
where equality is attained for every $d\geq 3$, and any curve where equality is attained lies on a quadric surface.

- Assembling facts about curves we know: 
  - For every $d\geq 1$ there is a smooth plane curve of degree $d$, all such have $g = {1\over 2}(d-1)(d-2)$.
  - For every $a,b\geq 1$ there is a complete intersection of surfaces in $\PP^3$ which is a smooth curve of degree $d = ab$ and genus $g = {1\over 2}ab(a+b-4)$.
  - For every $a,b\geq 1$ there are smooth curves of type $(a, b)$ on a smooth quadric surface, which have degree $d=a+b$ and $g=ab-a-b+1$.
  - If $X$ is a curve on a quadric cone $Q$: 
    - $d$ even $\implies X$ is a complete intersection of $Q$ with a surface of degree $a\da d/2$ with $g=a^2-2a+1$.
    - $d$ odd $\implies X$ has genus $g=a^2-a$ for $d \da 2a+1$.
- Classifying curves of degree 7: 
  - $d=1\implies X\cong \PP^1$.
  - $d=2\implies X$ is the conic in $\PP^2$.
  - $d=3\implies$
    - Plane cubic with $g=1$.
    - Twisted cubic in $\PP^3$ with $g=0$.
  - $d=4\implies$
    - Plane quartic with $g=3$
    - Rational quartic curves
    - Elliptic quartic curves, which are complete intersections of two quadric surfaces.
  - $d=5\implies$
    - Plane quintic with $g=6$
    - Curves with $\OO_X(1)$ nonspecial, with $g=0,1,2$
  - $d=6\implies$
    - Plane sextic with $g=10$
    - Curves with $\OO_X(1)$ nonspecial, with $g=0,1,2,3$
    - A curve with $g=4$, the complete intersection of a quadric surface and a cubic surface, the canonical curve for genus 4.
  - $d=7\implies$
    - Plane septic with $g=15$
    - Curves with $\OO_X(1)$ nonspecial, with $g=0,1,2,3,4$, and any $g\leq 4$ yields a nonspecial curve
    - A curve of type $(3,4)$ on a smooth quadric, with $g=6$.
    - $g=6$ curves, the maximum for $d=7$, all must lie on a quadric.

- Example of the complexities of classifying higher degrees/genera: there **are** curves with $(d, g) = (7,5)$.
  - Need a very ample $D$ of degree 7 with $h^0(D) \geq 3$; by RR $D$ is special.
  - Have $\deg K = 8$ so $D = K-P$ with $h^0(D) = 3$.
  - Very ample criterion: need $h^0(K-P-Q-R) = h^0(K-P) -2$, so $h^0(P+Q+R) = 0$ for all $Q,R$.
  - Only possible iff $X$ has no $g_3^1$, so $X$ embeds iff $X$ has no $g_{3}^1$, and such curves do exist.
- Summary of classification: 
  ![](figures/2023-02-04_18-32-41.png)

- Two distinct families of $(d,g) = (9, 10)$: 
  - Type 1: a complete intersection of cubic surfaces.
    - $\omega_X \cong \OO_X(2)$ so $\OO_X(2)$ is special and $h^0(\OO_X(2)) = 10$.
    - $X$ is projectively normal so $H^0(\OO_{\PP^3}(2))\surjects H^0(\OO_X(2))$.
    - SES implies $h^0(I_X(2)) = 0$, so $X$ is not contained in any quadric surface.
  - Type 2: a curve of type $(3, 6)$ on a smooth quadric surface $Q$.
    - Take $\OO_Q(-3, -6)\injects \OO_Q \surjects \OO_X$, twist by 2 and take LES to get $h^0(\OO_X(2)) = 9$ so $\OO(2)$ is nonspecial.
    - $X$ can not be contained in two distinct quadric surfaces, forcing $h^0(I_X(2)) = 1$.
  - Dimension of cohomology only *increases* under specialization, so by semicontinuity neither type is a specialization of the other: $h^0(I_X(2))$ increases from Type 1 to 2, and $h^0(\OO_X(2))$ decreases.
  - Any curve with $(d,g) = (9, 10)$ is one of these types: 
    - If $\OO_X(2)$ is nonspecial then $h^0(\OO(2)) = 9$ and $X$ is contained in some $Q$.
    - Possibilities for $(d,g)$ forces $Q$ to be smooth and $X$ to be of type $(3, 6)$.
    - If $\OO_X(2)$ is special then it can not lie on any $Q$: if it did, it would be type 2, making $\OO_X(2)$ nonspecial.
    - $\OO_X(3)$ is nonspecial since its degree is $\gt 2g-2$, so $h^0(\OO_X(3)) = 18$ and $h^0(I_X(2)) \geq 2$.
    - The corresponding cubics must be irreducible, so $X$ is contained in an intersection of two cubic surfaces $S_1$ and $S_2$.
    - Checking degrees for $X$ to be the complete intersection of $S_1, S_2$.
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