General degree $d$ surface in $\mathbb{P}^3$ - MathOverflow

General degree $d$ surface in $\mathbb{P}^3$

2012-08-04T18:04:19Z

Let $H_{d_1,g_1}, H_{d_2,g_2}$ be two Hilbert schemes of curves in $\mathbb{P}^3$ with degrees $d_1, d_2$ and genus $g_1, g_2$. Denote by $H:=H_{d_1,g_1}\times H_{d_2,g_2}$ where an element in $H$ is a pair of curves $(C_1, C_2)$ with $C_i \in H_{d_i,g_i}$.

As far as I understand, a generic element of $H$ consists of pairs $(C_1,C_2)$ such that $C_1 \cap C_2 = \emptyset$.

Assume now that $g_i \not=\frac{1}{2}(d-1)(d-2)$ i.e., the Hilbert scheme do not parametrize plane curves. My question is whether we can say something similar for a degree $d$ surface in $\mathbb{P}^3$. More precisely, for a fixed $d \ge 5$, we denote by $H'$ the space parametrizing the family of all smooth degree $d$ surfaces $X$ in $\mathbb{P}^3$ such that $X$ contains at least one pair of curves $(C_1, C_2) \in H$. Then can we say that a general $X$ in $H'$ contains a non-intersection pair of curves $(C_1, C_2) \in H$. This is equivalent to saying that the dimension of an irreducible component of $H'$ is maximal (among the components of $H'$) if it parametrizes surfaces containing $(C_1, C_2) \in H$ with $C_1 \cap C_2 = \emptyset$.

Note that we need to assume $C_i$ are not plane curves because as far as I recall it gives a counter-example to the statement.

Answer by Will Sawin for General degree $d$ surface in $\mathbb{P}^3$

2012-08-04T21:44:12Z

We can choose $d_1,g_1,d_2,g_2,d$ such that every pair of curves in $H$ contained in a single degree $d$ surface must intersect. The most well-behaved hypersurface that is not a plane is of course a quadric surface, so we choose $d=2$. A curve of degree $4$ and genus $1$ on a quadric surface must have bidegree $(2,2)$. Two such curves have intersection number $8$ and anyways must intersect. Clearly, they are not plane curves.

Answer by Jack Huizenga for General degree $d$ surface in $\mathbb{P}^3$

2012-08-04T22:05:27Z

By the Noether-Lefschetz theorem, a very general surface $X\subset \mathbb{P}^3$ of degree $d\geq 4$ has $\mathrm{Pic}(X) \cong \mathbb Z$, generated by a hyperplane section $H$. But then if $C \sim aH$ and $C'\sim bH$ on $X$, we have $C\cdot C' = abd^2 > 0$, and so $C,C'$ must intersect in at least one (and generically many) point. Thus every pair of curves on $X$ intersect each other.

(This doesn't rule out that for some choices of pairs of curve classes that what you want could be true, but it does rule out that the most general possible result is true).