## The Clemens Conjecture

This is a draft of a talk I’m giving on Thursday, mostly going over the work of Katz in this paper.

Let’s begin with an informal statement of the conjecture:

Conjecture: For every positive integer $d$, a general quintic hypersurface in $\mathbb{P}^4$ will contain only finitely many rational curves of degree $d$.

First off, we’ll justify WHY we should make such a conjecture:

Any rational curve can be viewed as being the image of $\phi:\mathbb{P}^1\to\mathbb{P}^4$ for some map. If the curve is of degree $d$, then the map $\phi$ consists of $\hphantom{}[\alpha_0(s,t):\ldots:\alpha_4(s,t)]$ with $\alpha_i$ homogeneous of degree $d$. Thus, each $\alpha_i$ has $d+1$ coefficients that we are free to choose, so the map has $5(d+1)=5d+5$ free parameters. However, $GL(2)$ acts on the space of maps, reparameterizing them, and this removes four degrees of freedom, so the space of curves is of dimension $5d+1$.

Now, to get constraints, let $F$ be a degree 5 polynomial on $\mathbb{P}^4$. A curve lies on $F$ iff $f(s,t):=F(\alpha_0(s,t),\ldots,\alpha_4(s,t))\equiv 0$. For this to happen, each coefficient of $f$ must be zero. Now, $f$ is of degree $5d$, and so $f$ has $5d+1$ coefficients which must vanish. This gives us the same number of constraints on the coefficients of the curve as there are dimensions in the space of curves, so we should hope to have only finitely many curves remaining.

Now we’ll state the conjecture rigorously:

Conjecture: Let $d>0$ be an integer. Then the scheme of smooth rational curves of degree $d$ on a generic quintic 3fold is finite, nonempty and reduced. Furthermore, each curve $C$ is embedded in the 3fold with normal bundle $\mathscr{O}_C(-1)\oplus \mathscr{O}_C(-1)$.

Our goal is, following Katz’s 1986 paper “On the finiteness of rational curves on quintic threefolds” to prove this conjecture for small $d$, specifically we’ll prove it for $d\leq 7$. To do this, we define the following:

• $\mathscr{M}_d=$the moduli space of smooth rational curves of degree $d$ in $\mathbb{P}^4$
• $P=\mathbb{P} H^0(\mathbb{P}^4,\mathscr{O}_{\mathbb{P}^4}(5))=$the space of quintics in $\mathbb{P}^4$
• $\mathscr{I}=\{(C,F)\in \mathscr{M}_d\times P|C\subset F\}$, the incidence scheme
• $\pi_1:\mathscr{I}\to \mathscr{M}_d$, $\pi_2:\mathscr{I}\to P$, the projection maps.

Now, here’s the main theorem:

Theorem (Katz): The Clemens conjecture holds for $d$ if

1. $\mathscr{I}$ is irreducible
2. There exists a curve $C$ which is smooth, rational, of degree $d$ on some smooth quintic threefold such that $N_{C/F}=\mathscr{O}_C(-1)\oplus \mathscr{O}_C(-1)$.

Proof: We start by defining $\mathscr{I}_0\subset \mathscr{I}$ to be the open subset where $F$ is smooth. Then for any $(C,F)\in \mathscr{I}_0$, we have the short exact sequence $0\to T_C\to T_F|_C\to N_{C/F}\to 0$. Noting that quintic threefolds are Calabi-Yau, and so have $K_F$ trivial and that $\deg T_C=2$, this tells us that $\deg N_{C/F}=-2$. Thus, either $N_{C/F}=\mathscr{O}_C(-1)\oplus \mathscr{O}_C(-1)$, or it has a global section.

Now, we look at $\mathscr{C}=\{(C,F,p)\in \mathscr{I}_0\times \mathbb{P}^4|p\in C\}$ and $\mathscr{F}=\{(C,F,p)\in \mathscr{I}_0\times\mathbb{P}^4|p\in F\}$, the “universal” curve and quintic respectively. Then $N_{\mathscr{C}/\mathscr{F}}$ is coherent and flat over $\mathscr{I}_0$, which tells us that $h^0(N_{C/F})$ is an upper semicontinuous function on $\mathscr{I}_0$!

This yields a stratification $\mathscr{I}_0\supset \mathscr{I}_1\supset\ldots$ where $\mathscr{I}_i=\{(C,F)\in \mathscr{I}_0|h^0(N_{C/F})\geq i\}$.

The second condition tells us that $\mathscr{I}_1$ is a proper subvariety, because there exists a curve with $h^0(N_{C/F})=0$.

Now, suppose that $\pi_2(\mathscr{I}_1)$ is dense in $P$. Then for generic $F$, we have $\pi^{-1}_2(F)\cap \mathscr{I}_1$ is a nonempty proper subvariety of $\pi_2^{-1}(F)\cap \mathscr{I}_0$. It will have positive codimension, because $\mathscr{I}_0$ is irreducible, by the first condition. So now we look at $\pi_2^{-1}(F)\cap \mathscr{I}_0\setminus \pi_2^{-1}(F)\cap \mathscr{I}_1$. This is a finite set, because any $(C,F)$ in it has $h^0(N_{C/F})=0$, and so has no deformations, which means there can’t be any continuous families here. But then, $\pi_2^{-1}(F)\cap \mathscr{I}_1$ is a proper subvariety of a finite set, and so is empty, which gives a contradiction with $\pi_2(\mathscr{I}_1)$ being dense in $P$.

So now, on the dense open set $P\setminus \overline{\pi_2(\mathscr{I}_1)}$, we have the finiteness we desired.

Now, the second condition along with teh fact that $\dim \mathscr{I}_0\geq \dim P$ tells us that we have a nonempty scheme of curves, adn it must be reduced because the tangent space is $H^0(N_{C/F})=0$.
QED.

So now we have reduced the Clemens conjecture to checking two things for each degree. The second condition holds for all $d$, due to a theorem of Mori: we obtain it by constructing rational curves of arbitrary degree on a quartic surface, and then the curve can be transported to a general quintic threefold via deformation theory.

The tricky part is showing that $\mathscr{I}$ is irreducible. We’re only going to manage it for $d\leq 7$. Before we do this, we’ll recall the following:

Definition: We call a coherent sheaf $\mathscr{F}$ on $\mathbb{P}^n$ $m$-regular if, $\forall i>0$, we have $H^i(\mathbb{P}^n,\mathscr{F}(m-i))=0$.

We say that a subvariety is $m$-regular if its ideal sheaf is.

It is a fact that if a sheaf is $m$-regular, then it is also $(m+1)$-regular.

With this definition, we cite a theorem

Theorem (Gruson, Lazarsfeld, Peskine, 1983): Let $C\subset \mathbb{P}^r$ be a reduced, irreducible, non-degenerate curve of degree $d$. Then $X$ is $(d+2-r)$-regular.

What we’re going to need are 6-regular curves. Before actually doing the rest of the proof, let’s spend a moment analyzing which curves we can be sure are 6-regular.

For nondegenerate curves in $\mathbb{P}^4$, we just get that a curve is $(d-2)$-regular, and so for nondegenerate curves, we can work up to degree 8. However, now let’s assume that the curve is degenerate, and falls into a $\mathbb{P}^3$. Then we have $(d+2-3)$-regularity, which means the curve is $(d-1)$-regular, and so here we can only work with curves up to degree 7, and there DO exist space octics on quintic 3folds, so we can’t prove that there are only finitely many octics. To be complete, we note that our requirement that curves be smooth gives us the degree-genus formula for plane curves, forcing our rational plane curves to be lines and conics only, and so, every smooth rational curve in $\mathbb{P}^4$ of degree $d\leq 7$, is 6-regular.

Now, how do we use 6-regularity? We first look at the short exact sequence $0\to \mathscr{I}_C\to \mathscr{O}_{\mathbb{P}^4}\to \mathscr{O}_C\to 0$. We then twist to get to $0\to \mathscr{I}_C(5)\to \mathscr{O}_{\mathbb{P}^4}(5)\to \mathscr{O}_C(5)\to 0$, and then take the long exact sequence on cohomology. This yields $0\to H^0(\mathscr{I}_C(5))\to H^0(\mathscr{O}_{\mathbb{P}^4}(5))\to H^0(\mathscr{O}_C(5))\to H^1(\mathscr{I}(6-1))=0$

with the last being due to $\mathscr{I}_C$ being 6-regular.

This tells us that the map $H^0(\mathscr{O}_{\mathbb{P}^4}(5))\to H^0(\mathscr{O}_C(5))$ is surjective. This tells us that we can find all of the quintics containing $C$ by looking at the ones that vanish on the polynomial, without having to worry about other quintic forms, which could confound our understanding of whether things are irreducible. Now, we have aspecific curve, so we have $\alpha_0,\ldots,\alpha_4$, and we plug these into the universal quintic on $\mathbb{P}^4$, getting $F(\alpha_0,\ldots,\alpha_4)$. As all the coefficients of this degree $5d$ polynomial must vanish, we get $5d+1$ linear conditions on the coefficients of the quintic. Thus, the fiber of $\pi_1$, that is, the set of quintics containing $C$, form a projective space of dimension $125-(5d+1)=124-5d$, and, in particular, this is irreducible. As $\pi_1$ maps $\mathscr{I}$ to an irreducible space with irreducible fibers, which yields the Clemens conjecture for $d\leq 7$.

In 1996, Johnsen and Kleiman were able to push this argument through to degrees 8 and 9 by a careful, case-by-case analysis of degree 8 and 9 curves. That same year, Johnsen and Kleiman made some progress on the cases 10-24, but didn’t obtain a complete proof of them. In 2005, Cotterill published a proof of the $d=10$ case, and in late 2007 posted a preprint for the $d=11$ case. As far as I know, this is the current state of knowledge on the conjecture.