# When is a del Pezzo surface a conic bundle?

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).

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,

• I was originally thinking k-birational to a conic bundle. This is still a helpful answer though
– fp1
Commented Mar 25 at 6:29
• I see. This looks a bit trickier. For example take a quartic del Pezzo surface $S$ with $\mathrm{Pic} S \cong \mathbb{Z}$. Such $S$ is minimal. If $S$ admits a general rational rational point, then blowing up it yields a cubic surface with a line, hence $S$ is birational to a conic bundle surface. If $S$ admits no rational points however, then it's not clear whether it is birational to a conic bundle surface of not. Commented Mar 25 at 11:52
• I'd be happy to hear more about how this question arose in your research; feel free to email me. Commented Mar 25 at 11:52