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I have an incidence variety $X = \{(p,\ell) \in C \times D^* : p \in \ell\} \subset \mathbb{P}^2 \times \mathbb{P}^2$, where $C = Z(f) \subset \mathbb{P}^2$ and $D^* = Z(g^*) \subset \mathbb{P}^2$ are two conics ($D^*$ is the dual conic of the conic $D = Z(g)$).

From Hartshorne exercise I.3.1(c) I know that every conic is isomorphic to $\mathbb{P}^1$. Can I use this to write $X$ as a variety in $\mathbb{P}^1 \times \mathbb{P}^1$? If so, what equations would define this variety?

Thanks in advance.

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    $\begingroup$ A hint: think about how many points of C are on a fixed line, and how many tangent lines (i.e. points of D^*) pass through a point of C. That will tell you the degrees of the equations which cut out X in P^1 x P^1, from which you can deduce various things such as the (arithmetic) genus of X. $\endgroup$ – user64687 Apr 23 '13 at 15:54
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    $\begingroup$ On the other hand, to get actual defining equations, you will need to pick explicit isomorphisms from C to P^1 and D* to P^1. $\endgroup$ – user64687 Apr 23 '13 at 15:54

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