TITLE: Volume of conic bundles
QUESTION [5 upvotes]: Consider a smooth conic bundle $X\rightarrow \mathbb{P}^1$ with discriminant of degree $d$ (the locus of $\mathbb{P}^1$ over which the fibers are reducible conics). There is a formula for $(-K_X)^2$ that reads $(-K_X)^2 = 8-d$. So that the volume depends only on the degree of the discriminant.
Now, consider conic bundles $Y\rightarrow\mathbb{P}^2$. For instance, if $Y\subset\mathbb{P}^2\times\mathbb{P}^2$ is a divisor of bidegree $(4,2)$ I got $(-K_Y)^3 = -6$. Note that in this case $d = 12$.
If $Y\subset\mathbb{P}(\mathcal{O}_{\mathbb{P}^2}(2)\oplus \mathcal{O}_{\mathbb{P}^2}(1)\oplus \mathcal{O}_{\mathbb{P}^2})$ given by an equation of the form
$$
a_0x^2+a_1xy+a_2xz+a_3y^2+a_4yz+a_5z^2 = 0
$$
where the $a_i$ are polynomials on the base $\mathbb{P}^2$ of degree respectively $6,5,4,4,3,2$. Note that also in this case $d = 12$.
So it seems that unlike the case of surfaces in higher dimension the is no formula for $(-K_Y)^n$ where $n$ is the dimension of the conic bundle $Y$.
Is this correct and if so is there a conceptual explanation for this fact?

REPLY [3 votes]: If I understand your question, then there is no such formula. Since there are examples where the degree of the discriminant curve is $5$ but with different $(-K_{X})^3$.
The blow up of a smooth cubic $3$-fold in a line is a conic bundle over $\mathbb{P}^2$ with discriminant curve of degree $5$. It is a Fano $3$-fold with $(-K)^3 =18$.
On the other hand Panin showed that the blow-up of $\mathbb{P}^3$ in a smooth curve of degree $7$ and genus $5$ is a conic bundle over $\mathbb{P}^2$ with  discriminant curve of degree $5$. It is a Fano $3$-fold, it has $(-K)^3 =16$.
I learned both these examples from https://arxiv.org/pdf/1712.05564.pdf, (example 3.4.2 and example 3.4.3 respectively). There is many more examples of conic bundles over $\mathbb{P}^2$ given there, and I guess many more counter examples to the formula.