In this paper we study the birational rigidity problem for smooth Mori fibrations on del Pezzo surfaces of degree 1 and 2. For degree 1 we obtain a complete description of rigid and non-rigid cases. 1. Notions and results. Due to progress in the Minimal Model Program, the classification problem in the modern birational geometry for varieties of negative Kodaira dimension can be formulated as fo...

We address the question: how common is it for a 3–manifold to fiber over the circle? One motivation for considering this is to give insight into the fairly inscrutable Virtual Fibration Conjecture. For the special class of 3–manifolds with tunnel number one, we provide compelling theoretical and experimental evidence that fibering is a very rare property. Indeed, in various precise senses it ha...

We construct a class of geometric twists for Calabi-Yau manifolds of Voisin-Borcea type (K3 × T )/Z2 and study the superpotential in a type IIA orientifold based on this geometry. The twists modify the direct product by fibering the K3 over T 2 while preserving the Z2 involution. As an important application, the Voisin-Borcea class contains T /(Z2 × Z2), the usual setting for intersecting D6 br...

Abstract In this paper, we study multiplicity results for double phase problems of Kirchhoff type with right-hand sides that include a parametric singular term and nonlinear subcritical growth. Under very general assumptions on the data, prove existence at least two weak solutions have different energy sign. Our treatment is based fibering method in form Nehari manifold. We point out cover both...

An [Formula: see text]-dimensional closed flat manifold is said to be of diagonal type if the standard representation its holonomy group text] diagonal. Bieberbach fundamental such a manifold. We introduce Vasquez invariant as least integer that every with fibers over dimension at most torus fibers. Using combinatorial description groups type, we give both upper and lower bounds for this invari...

In this paper we apply the fibering method of Pohozaev and notion extremal values introduced by Il’yasov to a Schrödinger—Poisson system, with prescribed L2 norm unknown, in whole ?3. The makes clear role played special exponents p = 3, 8/3, 10/3. addition showing that old results can be obtained unified way, exhibit also new ones.