Assume we are given a nontrivial standard conic bundle $\pi: X\rightarrow S$, that is $X$ and $S$ are smooth projective varieties (say over $\mathbb{C}$), $\pi$ is flat and furthermore we have $Pic(X)=\pi^{*}Pic(S)\oplus \mathbb{Z}K_X$.

In [1, Proposition 2.4] it is proved, that if $S$ is a smooth projective surface and $\sigma: S_1\rightarrow S$ is the blow up at a point $p\in S$, then $\pi_1:X_1\rightarrow S_1$ is a standard conic bundle, here we have $X_1:=X\times_S S_1$. This is done by an analysis of the position of $p$ relative to the discriminant curve $D$, that is one considers the cases $p\notin D$, $p\in D\backslash sing(D)$ and $p\in sing(D)$. In the first case the discriminat on $S_1$ is $\sigma^{-1}(D)$ but in the other two cases it is $\sigma^{-1}(D)\backslash\sigma^{-1}(p)$.

$\underline{Question}$: What happens in higher dimension, i.e. $dim(S)>2$? Can we describe the (or some) smooth irreducible centers $Z\subset S$ such that the blow up $\sigma_Z: S_Z \rightarrow S$ of $Z$ induces a standard conic bundle $\pi_Z: X_Z\rightarrow S_Z$, for $X_Z=X\times_S S_Z$. How does the discriminant curve look like in $S_Z$?

In [2, Lemma 2.2] it is stated, that if $\pi: X\rightarrow S$ is a standard conic bundle with discriminant divisor $D$ and $Z\cap D=\emptyset$, then $\pi_Z: X_Z\rightarrow S_Z$ is a standard conic bundle, with discriminant divisor being $\sigma_Z^{-1}(D)$. There is also a statement about the blow up at a center $Z\subset D$, having the property that the restriction of the double cover of $D$ defined by $\pi$ to $Z$ splits. But this only gives, as far as I understand, a standard conic birationally equivalent to $X_Z$ but does not have to be actually $X_Z$.

Maybe already the case $dim(S)=3$ is of interest. Here the possibilities for $Z$ are a point or a smooth irreducible curve. How do they need to lie relatively to $D$ such that the blow up of $Z$ produces a standard conic bundle? I am not an expert when it comes to blow ups in higher dimension, but I think the main problem will be ensuring that $X_Z$ is smooth. In [1] this is done by some nice geometric arguments, can this be done in higher dimension also?

[1] Sarkisov - Birational automorphisms of conic bundles

[2] Sarkisov - On conic bundle structures


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.