Narrow Results

Let $𝕜$ be any field of characteristic zero, $X$ be a del Pezzo surface and $G$ be a finite subgroup in $Aut(X)$. In this paper, we study when the quotient surface $X/G$ can be non-rational over $𝕜$. Obviously, if there are no smooth $𝕜$-points on $X/G$ then it is not $𝕜$-rational. Therefore, under assumption that the set of smooth $𝕜$-points on $X/G$ is not empty we show that there are few possibilities for non-$𝕜$-rational quotients.

The quotients of del Pezzo surfaces of degree $2$ and greater are considered in the author’s previous papers. In this paper, we study the quotients of del Pezzo surfaces of degree $1$. We show that they can be non-$𝕜$-rational only for the trivial group or cyclic groups of order $2$, $3$ and $6$. For the trivial group and the group of order $2$, we show that both $X$ and $X/G$ are not $𝕜$-rational if the $G$-invariant Picard number of $X$ is $1$. For the groups of order $3$ and $6$, we construct examples of both $𝕜$-rational and non-$𝕜$-rational quotients of both $𝕜$-rational and non-$𝕜$-rational del Pezzo surfaces of degree $1$ such that the $G$-invariant Picard number of $X$ is $1$.

As a result of complete classification of non-$𝕜$-rational quotients of del Pezzo surfaces we classify surfaces that are birationally equivalent to quotients of $𝕜$-rational surfaces, and obtain some corollaries concerning fields of invariants of $𝕜(x,y)$.

We introduce the notion of strongly asymptotically log del Pezzo flags, and classify such flags under the assumption that their zero-dimensional part lies in the boundary. We use this result to give a new and conceptual proof of the classification of strongly asymptotically log del Pezzo surfaces, originally due to Cheltsov and Rubinstein.

We establish a measure which describes precisely the local asymptotic distribution of rational points outside the locally accumulating subvarieties around a general rational point on a del Pezzo surface of degree 6 in the sense of diophantine approximation.

We determine restrictions on the singularity content of a Fano polygon, or equivalently of certain orbifold del Pezzo surfaces. We establish bounds on the maximum number of $\frac{1}{R}\left(1,\text{}\text{}1\right)$ singularities in the basket of residual singularities. In particular, there are no Fano polygons without T-singularities and with a basket given by (i) $\left\{k\times \frac{1}{R}(1,1)\right\}$ for k ∈ ℤ_>0 and R ≥ 5, or (ii) $\left\{\frac{1}{R_1}(1,1),\frac{1}{R_2}(1,1),\frac{1}{R_3}(1,1)\right\}$.