Does there exists a smooth projective surface $X$ which contains a projective line $L$ and a smooth conic $C$ such that $L\cap C=$empty?

up vote 4 down vote accepted

Yes. This holds for any cubic surface over an algebraically closed field $k$.

Let $S$ be such a surface. Let $L'$ be a line. The pencil of hyperplanes containing $L'$ forms a conic bundle $\pi: S \to \mathbb{P}^1$. This conic bundle has $5$ singular fibres, which are each a pair of lines meeting in a point. Take $C$ a smooth fibre and $L$ one of the lines in a singular fibre. Then $L \cap C = \emptyset$.

Consider a smooth cubic surface $X$, namely the blow up of $\mathbb{P}^2_k$ at six distinct points $\{p_1, \ldots, p_6 \}$ in general position.

Then you can take as $C$ the strict transform in $X$ of a conic through $4$ of the points $p_i$ and as $L$ the exceptional divisor at one of the remaining points.

Yet another example is a cubic scroll. Let $L,C \subset \mathbb{P}^4$ be a line and a conic, whose linear span does not intersect the line. Choose an isomorphism $L \cong \mathbb{P}^1 \cong C$ and let $S$ be the union of lines in $\mathbb{P}^4$ joining points in $L$ with their images in $C$. Alternatively, $$ S \cong \mathbb{P}_{\mathbb{P}^1}(O(1) \oplus O(2)). $$

Your Answer

 
discard

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.