Related papers: Lagrangian spheres in Del Pezzo surfaces

We prove that the only exact Lagrangian submanifolds in an ALE space are spheres. ALE spaces are the simply connected hyperkahler manifolds which at infinity look like C^2/G for any finite subgroup G of SL(2,C). They can be realized as the…

We examine static spherically symmetric polytropic spheres in Palatini f(R) gravity and show that no regular solutions to the field equations exist for physically relevant cases such as a monatomic isentropic gas or a degenerate electron…

The correspondence between del Pezzo surfaces and field theory models over the complex numbers or for split real forms is extended to other real forms, in particular to those compatible with supersymmetry. Specifically, all theories of the…

We give examples of Calabi-Yau threefolds containing Lagrangian spheres which are not vanishing cycles of nodal degenerations, answering a question of Donaldson in the negative.

In this article, we study the singularities of Lagrangian immersions into Cartesian product of surfaces. After applying a Hamiltonian isotopy in the Weinstein tubular neighbourhood of the Lagrangian immersion, the singular points of the…

We introduce two basic invariant forms which define generic surface in 3-space uniquely up to Lie sphere equivalence. Two particularly interesting classes of surfaces associated with these invariants are considered, namely, the Lie-minimal…

We show that, up to Lagrangian isotopy, there is a unique Lagrangian torus inside each of the following uniruled symplectic four-manifolds: the symplectic vector space $\mathbb{R}^4$, the projective plane $\mathbb{C}P^2$, and the monotone…

We state conditions under which the set S(k) of k-rational points on a del Pezzo surface S of degree 1 over an infinite field k of characteristic not equal to 2 or 3 is Zariski dense. For example, it suffices to require that the elliptic…

The complex projective plane CP^2 contains certain Lagrangian CW-complexes called pinwheels, which have interesting rigidity properties related to solutions of the Markov equation. We compute the Gromov width of the complement of pinwheels…

Using the degeneration technique, one studies the behavior of Welschinger invariants under the blow-up, and obtains some blow-up formulae of Welschinger invariants. One also analyses the variation of Welschinger invariants when replacing a…

We classify quartic del Pezzo surface fibrations over the projective line via numerical invariants, giving explicit examples for small values of the invariants. For generic such fibrations, we describe explicitly the geometry of spaces of…

We classify the topological types of surfaces in the 3-dimensional unit sphere that contain both a great and a small circle through each point. In particular, these surfaces are homeomorphic to one of five normal forms and are either the…

We study the formation of clusters of passive Lagrangian tracers in a non-smooth turbulent flow in a flat free-slip surface as a model for particle dynamics on free surfaces. Single particle and pair dispersion show different behavior for…

It is well known that every Del Pezzo surface of degree 5 defined over k is parametrizable over k. In this paper we give an efficient construction for parametrizing, as well as algorithms for constructing examples in every isomorphism class…

This is the first article in a series aimed at classifying normal del Pezzo surfaces of Picard rank one over algebraically closed fields of arbitrary characteristic up to an isomorphism. Our guiding invariant is the height of a del Pezzo…

We consider a family of closed symplectic manifolds 4-manifolds which we call symplectic bielliptic surfaces and study its Lagrangian cobordism group of weakly-exact Lagrangian G-branes (that is, Lagrangians equipped with a grading, a Pin…

Let k be a perfect field. Recently J.-L. Colliot-Th\'el\`ene showed that two pointless quadric surfaces over k are birationally equivalent if and only if they are isomorphic. We show that this result holds for arbitrary del Pezzo surfaces…

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…

We classify Galois actions on Picard lattices of del Pezzo surfaces of degrees 1,2, and 3 giving rise to minimal surfaces with no cohomological obstructions to stable rationality.

In this article we explore compactifications of cluster varieties of finite type in complex dimension two. Cluster varieties can be viewed as the spec of a ring generated by theta functions and a compactification of such varieties can be…