When is a del Pezzo surface a conic bundle?

Question

I am considering over a field $k$ which is not algebraically closed, characteristic 0, and perhaps contains all the complex roots of unity that may appear. Feel free to realize it as some function fields such as $\mathbb{C}(x,y)$.

Given $X$ a del Pezzo surface of degree 2 (or any degree if there's a more general theorem), it is known that $X$ may be $k$-birational to a conic bundle with 6 (or 8-deg) degenerate fibers. It is also known that if $X$ is indeed a conic bundle then the Picard group has rank $\geq 2$. Do we have an iff theorem for determining a given del Pezzo surface is indeed a conic bundle? By "given" I mean for example an explicit equation $w^2=f_4(u,v,t)$.

I would also appreciate some explicit examples on this. I know Swinnerton-Dyer has studied conic bundles $r^2+s^2=f_2(t)f_4(t)$, but I don't think he gave example of del Pezzo surfaces birational to this form (I think he also worked over $\mathbb{Q}$ in stead of a function field).

Answer 1

It is not clear from the question whether you want to determine whether $X$ admits a conic bundle structure, or whether it is birational to a conic bundle surface. I will focus on the former. If you are not already familiar with the work of Iskovskih [1], I would recommend to study this paper as he says a bit about minimal models.

Any del Pezzo surface of degree two with a conic bundle structure can be embedded as a surface of bidegree $(2,2)$ in $\mathbb{P}^1 \times \mathbb{P}^2$. This map to $\mathbb{P}^1 \times \mathbb{P}^2$ is given by simply the conic bundle morphism and the anticanonical map.

However given a del Pezzo surface of degree 2 with equation in the weighted projective space $\mathbb{P}(1,1,1,2)$ it can be tricky to determine whether it has a conic bundle bundle. This turns out to being equivalent to there being a Galois invariant collection of $(-1)$-curves which look like some singular fibres of a conic bundle structure (for example two Galois conjugate $(-1)$-curves meeting in a single point). For del Pezzo surfaces of degree 2 the $(-1)$-curves can be theoretically calculated through the bitangents to the corresponding plane quartic curve.

You can read more about some of this in the paper [2]. See in particular Section 5, and Proposition 5.2 is the criterion I mention above.

[1] Iskovskih, V. A. Minimal models of rational surfaces over arbitrary fields. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 19--43, 237.

[2] Frei, Christopher; Loughran, Daniel; Sofos, Efthymios. Rational points of bounded height on general conic bundle surfaces. Proc. Lond. Math. Soc. (3) 117 (2018), no. 2,