Related papers: Real del Pezzo surfaces without points

We compute the complexity of del Pezzo surfaces with du Val singularities.

We obtain a formula for the number of genus two curves with a fixed complex structure of a given degree on a del-Pezzo surface that pass through an appropriate number of generic points of the surface. This is done by extending the…

We study the arithmetic of del Pezzo surfaces $Y$ of degree 2 over a function field, and in particular, the cokernel of the homomorphism from the Picard group to the Galois-invariants of the geometric Picard group $\operatorname{Pic} Y…

We prove that there is a unique $R$-equivalence class on every del Pezzo surface of degree $4$ defined over the Laurent field $K=k((t))$ in one variable over an algebraically closed field $k$ of characteristic not equal to $2$ or $5$. We…

We study the birational properties of geometrically rational surfaces from a derived categorical point of view. In particular, we give a criterion for the rationality of a del Pezzo surface over an arbitrary field, namely, that its derived…

In these notes, we consider self-maps of degree > 1 on a weak del Pezzo surface X of degree < 8. We show that there are exactly 12 such X, modulo isomorphism. In particular, K_X^2 > 2, and if X has one self-map of degree > 1 then for every…

We present efficient classification algorithms for log del Pezzo surfaces with torus action of Picard number one and given Gorenstein index. Explicit results are obtained up to Gorenstein index 200.

We prove Manin's conjecture for split smooth quintic del Pezzo surfaces over arbitrary number fields with respect to fairly general anticanonical height functions. After passing to universal torsors, we first show that we may restrict the…

Over an algebraically closed field k, there are 16 lines on a degree 4 del Pezzo surface, but for other fields the situation is more subtle. In order to improve enumerative results over perfect fields, Kass and Wickelgren introduce a method…

We describe the effect of rational singularities on the Brauer group of a surface, and compute the Brauer groups of all singular del Pezzo surfaces over an algebraically closed field.

In this paper, we consider the following problem: Does there exist a cubic surface over $\mathbb{Q}$ which contains no line over $\mathbb{Q}$, yet contains a line over every completion of $\mathbb{Q}$? This question may be interpreted as…

Let X be a smooth, complex Fano 4-fold, and rho(X) its Picard number. We show that if rho(X)>12, then X is a product of del Pezzo surfaces. The proof relies on a careful study of divisorial elementary contractions f: X->Y such that the…

Let $K$ be a number field and $S$ a finite set of primes of $K$. Scholl proved that there are only finitely many $K$-isomorphism classes of del Pezzo surfaces of any degree $1 \le d \le 9$ over $K$ with good reduction away from $S$. Let…

We classify the number of $k$-rational lines and conic fibrations on del Pezzo surfaces over a field $k$ in terms of relatively minimal surfaces and establish rational curve analogues of the inverse Galois problem for del Pezzo surfaces. We…

The Manin-Peyre conjecture is established for a split singular quintic del Pezzo surface with singularity type $\mathbf{A}_2$ and two split singular quartic del Pezzo surfaces with singularity types $\mathbf{A}_3+\mathbf{A}_1$ and…

We classify the automorphism groups of del Pezzo surfaces of degrees one and two over an algebraically closed field of characteristic two. This finishes the classification of automorphism groups of del Pezzo surfaces in all characteristics.

This is the fourth of a series of papers studying real algebraic threefolds, but the methods are mostly independent from the previous ones. Let $f:X\to C$ be a map of a smooth projective real algebraic 3-fold to a curve $C$ whose general…

Using Fedder's criterion, we classify all non-$F$-split del Pezzo surfaces of degree $1$. We give a necessary and sufficient criterion for the $F$-splitting of such del Pezzo surfaces in terms of their anti-canonical system.

In this paper, we classify Du Val del Pezzo surfaces of Picard rank one in characteristic two and three. We also show that if a Du Val del Pezzo surface is Frobenius split, then a general anti-canonical member is smooth. Furthermore, in…

Let X be a real algebraic surface. The comparison between the volume of real and complex loci of ample divisors D brings us to define the concordance, which is a number between 0 and 1. This number equals 1 when the Picard number is 1, and…