An overview of Manin's conjecture for del Pezzo surfaces

Manin's conjecture is proved for a split del Pezzo surface of degree 5 with a singularity of type A_2.

We show that, for every integer $1 \leq d \leq 4$ and every finite set $S$ of places, there exists a degree $d$ del Pezzo surface $X$ over ${\mathbb Q}$ such that ${\rm Br}(X)/{\rm Br}({\mathbb Q}) \cong {\mathbb Z}/2{\mathbb Z}$ and the Brauer-Manin obstruction works exactly at the places in $S$. For $d = 4$, we prove that in all cases, with the exception of $S = \{\infty\}$, this surface may be chosen diagonalizably over ${\mathbb Q}$.

We give upper bounds for the number of rational points of bounded anti-canonical height on del Pezzo surfaces of degree at most five over any global field whose characteristic is not equal to two or three. For number fields these results are conditional on a conjecture relating the rank of an elliptic curve to its conductor, while they are unconditional in positive characteristic. For quartic or quintic del Pezzo surfaces with a conic bundle structure, we establish even stron...

We study singular del Pezzo surfaces that are quasi-smooth and well-formed weighted hypersurfaces. We give an algorithm how to classify all of them.

This paper focuses on the classification of all toric log Del Pezzo surfaces with exactly one singularity up to isomorphism, and on the description of how they are embedded as intersections of finitely many quadrics into suitable projective spaces.

As another application of the degeneration methods of [V3], we count the number of irreducible degree $d$ geometric genus $g$ plane curves, with fixed multiple points on a conic $E$, not containing $E$, through an appropriate number of general points in the plane. As a special case, we count the number of irreducible genus $g$ curves in any divisor class $D$ on the blow-up of the plane at up to five points (no three collinear). We then show that these numbers give the genus $...

In his book "Cubic forms" Manin discovered that del Pezzo surfaces are related to root systems. To explain the many numerical coincidences Batyrev conjectured that a universal torsor on a del Pezzo surface can be embedded in a certain projective homogeneous space of the semisimple group with the same root system, equivariantly with respect to the maximal torus action. We prove this conjecture for del Pezzo surfaces of degrees greater than 1. Our proof uses an inductive proces...

We report on our investigations concerning algebraic and transcendental Brauer-Manin obstructions to integral points on complements of a hyperplane section in degree four del Pezzo surfaces. We discuss moreover two concepts of an obstruction at an archimedean place. Concrete examples are given of pairs of non-homogeneous quadratic polynomials in four variables representing (0,0) over Q and over Z_p for all primes p, but not over Z. By blow-up, these yield cubic polynomials in...

We develop a strategy to classify the components of the space of sections of a del Pezzo fibration over $\mathbb{P}^{1}$. In particular, we prove the Movable Bend and Break lemma for del Pezzo fibrations. Our approach is motivated by Geometric Manin's Conjecture and proves upper bounds on the associated counting function. We also give applications to enumerativity of Gromov-Witten invariants and to the study of the Abel-Jacobi map.

We classify all log del Pezzo surfaces of Picard number one defined over algebraically closed fields of characteristic different from two and three. We also discuss some consequences of the classification. For example, we show that log del Pezzo surfaces of Picard number one defined over algebraically closed fields of characteristic higher than five admit a log resolution that lifts to characteristic zero over a smooth base.