Del Pezzo Surface Of Degree Research Articles

Weak weak approximation and the Hilbert property for degree 2 del Pezzo surfaces

AbstractWe prove that del Pezzo surfaces of degree 2 over a field satisfy weak weak approximation if is a number field and the Hilbert property if is Hilbertian of characteristic zero, provided that they contain a ‐rational point lying neither on any 4 of the 56 exceptional curves nor on the ramification divisor of the anticanonical morphism. This builds upon results of Manin, Salgado–Testa–Várilly‐Alvarado, and Festi–van Luijk on the unirationality of such surfaces, and upon work of the first two authors verifying weak weak approximation under the assumption of a conic fibration.

On local stability threshold of del Pezzo surfaces

AbstractWe complete the classification of local stability thresholds for smooth del Pezzo surfaces of degree 2. In particular, we show that this number is irrational if and only if there is a unique (‐1)‐curve passing through the point where we are computing the local invariant.

Frobenius actions on Del Pezzo surfaces of degree 2

Frobenius actions on Del Pezzo surfaces of degree 2

The 3-dimensional Lyness map and a self-mirror log Calabi–Yau 3-fold

AbstractThe 2-dimensional Lyness map is a 5-periodic birational map of the plane which may famously be resolved to give an automorphism of a log Calabi–Yau surface, given by the complement of an anticanonical pentagon of $$(-1)$$ ( - 1 ) -curves in a del Pezzo surface of degree 5. This surface has many remarkable properties and, in particular, it is mirror to itself. We construct the 3-dimensional big brother of this surface by considering the 3-dimensional Lyness map, which is an 8-periodic birational map. The variety we obtain is a special (non-$${\mathbb {Q}}$$ Q -factorial) affine Fano 3-fold of type $$V_{12}$$ V 12 , and we show that it is a self-mirror log Calabi–Yau 3-fold.

A Zariski dense exceptional set in Manin’s Conjecture: dimension 2

Recently, Lehmann, Sengupta, and Tanimoto proposed a conjectural construction of the exceptional set in Manin’s Conjecture, which we call the geometric exceptional set. We construct a del Pezzo surface of degree 1 whose geometric exceptional set is Zariski dense. In particular, this provides the first counterexample to the original version of Manin’s Conjecture in dimension 2 in characteristic 0. Assuming the finiteness of Tate-Shafarevich groups of elliptic curves over $${\mathbb Q}$$ with j-invariant 0, we show that there are infinitely many such counterexamples.

On wall-crossing invariance of certain sums of Welschinger numbers

We continue our quest for real enumerative invariants not sensitive to changing the real structure and extend the construction we uncovered previously for counting curves of anti-canonical degree $$\leqslant 2$$ on del Pezzo surfaces with $$K^2=1$$ to curves of any anti-canonical degree and on any del Pezzo surfaces of degree $$K^2\leqslant 3$$ .

Twistors, Quartics, and del Pezzo Fibrations

It has been known that twistor spaces associated to self-dual metrics on compact 4-manifolds are source of interesting examples of non-projective Moishezon threefolds. In this paper we investigate the structure of a variety of new Moishezon twistor spaces. The anti-canonical line bundle on any twistor space admits a canonical half, and we analyze the structure of twistor spaces by using the pluri-half-anti-canonical map from the twistor spaces. Specifically, each of the present twistor spaces is bimeromorphic to a double covering of a scroll of planes over a rational normal curve, and the branch divisor of the double cover is a cut of the scroll by a quartic hypersurface. In particular, the double covering has a pencil of Del Pezzo surfaces of degree two. Correspondingly, the twistor spaces have a pencil of rational surfaces with big anti-canonical class. The base locus of the last pencil is a cycle of rational curves, and it is an anti-canonical curve on smooth members of the pencil. These twistor spaces are naturally classified into four types according to the type of singularities of the branch divisor, or equivalently, those of the Del Pezzo surfaces in the pencil. We also show that the quartic hypersurface satisfies a strong constraint and as a result the defining polynomial of the quartic hypersurface has to be of a specific form. Together with our previous result in Honda (“A new series of compact minitwistor spaces and Moishezon twistor spaces over them”, 2010), the present result completes a classification of Moishezon twistor spaces whose half-anti-canonical system is a pencil. Twistor spaces whose half-anti-canonical system is larger than pencil have been understood for a long time before. In the opposite direction, no example is known of a Moishezon twistor space whose half-anti-canonical system is smaller than a pencil. Twistor spaces which have a similar structure were studied in Honda (“Double solid twistor spaces: the case of arbitrary signature”, 2008 and “Double solid twistor spaces II: General case”, 2015) and they are very special examples among the present twistor spaces.

Birational classification of pointless del Pezzo surfaces of degree 8

Let $$\Bbbk $$ be a perfect field. Recently Jean-Louis Colliot-Thélène showed that two pointless quadric surfaces over $$\Bbbk $$ are birationally equivalent if and only if they are isomorphic. We show that this result holds for arbitrary del Pezzo surfaces of degree 8 with the Picard number 1, and describe minimal surfaces birationally equivalent to a given pointless del Pezzo surface of degree 8.

Forms of del Pezzo surfaces of degree $5$ and $6$

We obtain necessary and sufficient condition for the existence of del Pezzo surfaces of degrees $5$ and $6$ over a field $K$ with a prescribed action of absolute Galois group $\operatorname{Gal}(K^{\mathrm{sep}}/K)$ on the graph of $(-1)$-curves. We also compute the automorphism groups of del Pezzo surfaces of degree $5$ over arbitrary fields. Bibliography: 19 titles.

Forms of del Pezzo surfaces of degree $5$ and $6$

В этой работе получено необходимое и достаточное условие для существования поверхностей дель Пеццо степеней 5 и 6 над полем $K$ с заданным действием абсолютной группы Галуа $\operatorname{Gal} ( K^{\mathrm{sep}}/K )$ на графе $(-1)$-кривых. Также вычислены группы автоморфизмов поверхностей дель Пеццо степени 5 над произвольными полями. Библиография: 19 названий.

Cohomology of moduli spaces of Del Pezzo surfaces

AbstractWe compute the rational Betti cohomology groups of the coarse moduli spaces of geometrically marked Del Pezzo surfaces of degree 3 and 4 as representations of the Weyl groups of the corresponding root systems. The proof uses a blend of methods from point counting over finite fields and techniques from arrangement complements.

Classifying sections of del Pezzo fibrations, II

Let $X$ be a del Pezzo surface over the function field of a complex curve. We study the behavior of rational points on $X$ leading to bounds on the counting function in Geometric Manin's Conjecture. A key tool is the Movable Bend and Break Lemma which yields an inductive approach to classifying relatively free sections for a del Pezzo fibration over a curve. Using this lemma we prove Geometric Manin's Conjecture for certain split del Pezzo surfaces of degree $\geq 2$ admitting a birational morphism to $\mathbb P^2$ over the ground field.

Concurrent lines on del Pezzo surfaces of degree one

Let X X be a del Pezzo surface of degree one over an algebraically closed field, and K X K_X its canonical divisor. The morphism φ \varphi induced by | − 2 K X | |-2K_X| realizes X X as a double cover of a cone in P 3 \mathbb {P}^3 , ramified over a smooth sextic curve. The surface X X contains 240 exceptional curves. We prove the following statements. For a point P P on the ramification curve of φ \varphi , at most sixteen exceptional curves contain P P in characteristic 2 2 , and at most ten in all other characteristics. Moreover, for a point Q Q outside the ramification curve, at most twelve exceptional curves contain Q Q in characteristic 3 3 , and at most ten in all other characteristics. We show that these upper bounds are sharp, except possibly in characteristic 5 outside the ramification curve.

The discriminant of a hypersurface in weighted projective space

In this paper, we study the discriminant of a hypersurface in a weighted projective space with isolated singularities. We define the discriminant of a hypersurface in [Formula: see text] with [Formula: see text], and show that it satisfies a smoothness criterion over any commutative ring. A smooth hypersurface of degree [Formula: see text] in the weighted projective three-fold [Formula: see text] is a del Pezzo surface of degree [Formula: see text]. We show that the determinant of the Galois action on the [Formula: see text]-lattice arising from its exceptional curves can be computed via the square roots of the discriminant.

Bounds for del pezzo surfaces of degree two

Abstract In this article, we obtain an upper bound for the number of integral points on del Pezzo surfaces of degree two.

Rational points on Del Pezzo surfaces of degree four

We study the distribution of the Brauer group and the frequency of the Brauer–Manin obstruction to the Hasse principle and weak approximation in a family of smooth del Pezzo surfaces of degree four over the rationals.

Weak Approximation for del Pezzo Surfaces of Low Degree

Abstract We prove, via an “arithmetic surjectivity” approach inspired by work of Denef, that weak weak approximation holds for surfaces with two conic fibrations satisfying a general assumption. In particular, weak weak approximation holds for general del Pezzo surfaces of degrees $1$ or $2$ with a conic fibration.

Density of rational points on a family of del Pezzo surfaces of degree one

Let k be an infinite field of characteristic 0, and X a del Pezzo surface of degree d with at least one k-rational point. Various methods from algebraic geometry and arithmetic statistics have shown the Zariski density of the set X(k) of k-rational points in X for d≥2 (under an extra condition for d=2), but fail to work in generality when the degree of X is 1, leaving a large class of del Pezzo surfaces for which the question of density of rational points is still open. In this paper, we prove the Zariski density of X(k) when X has degree 1 and is represented in the weighted projective space P(2,3,1,1) with coordinates x,y,z,w by an equation of the form y2=x3+az6+bz3w3+cw6 for a,b,c∈k with a,c non-zero, under the condition that the elliptic surface obtained by blowing up the base point of the anticanonical linear system |−KX| contains a smooth fiber above a point in P1∖{(1:0),(0:1)} with positive rank over k. When k is of finite type over Q, this condition is sufficient and necessary.

Unirationality of RDP Del Pezzo surfaces of degree 2

We study unirationality of a Del Pezzo surface of degree two over a given (non algebraically closed) field, under the assumption that it admits at least one rational double point over an algebraic closure of the base field. As corollaries of our main results, we find that over a finite field, it is unirational if the cardinality of the field is greater than or equal to nine and we also find that over an infinite field, which is not necessarily perfect, it is unirational if and only if the rational points are Zariski dense over the field.

Which rational double points occur on del Pezzo surfaces?

We determine all configurations of rational double points that occur on RDP del Pezzo surfaces of arbitrary degree and Picard rank over an algebraically closed field $k$ of arbitrary characteristic ${\rm char}(k)=p \geq 0$, generalizing classical work of Du Val to positive characteristic. Moreover, we give simplified equations for all RDP del Pezzo surfaces of degree $1$ containing non-taut rational double points.