We study del Pezzo surfaces that are quasismooth and well-formed weighted hypersurfaces. In particular, we find all such surfaces whose α-invariant of Tian is greater than 2/3.
Let ϕ: X → X be a morphism of a variety over a number field K. We consider local conditions and a “Brauer–Manin” condition, defined by Hsia and Silverman, for the orbit of a point P ∈ X(K) to be disjoint from a subvariety V⊆X, i.e., for We provide evidence that the dynamical Brauer–Manin condition is sufficient to explain the lack of points in the intersection ; this evidence stems from a probabilistic argument as well as unconditional results in the case of étale maps.
We give the first examples of smooth Fano and Calabi–Yau varieties violating the (narrow) canonical strip hypothesis, which concerns the location of the roots of Hilbert polynomials of polarized varieties. They are given by moduli spaces of rank 2 bundles with fixed odd-degree determinant on curves of sufficiently high genus, hence our Fano examples have Picard rank 1, index 2, are rational, and have moduli. The hypotheses also fail for several other closely related varieties.
We conjecture that the stable Khovanov homology of torus knots can be described as the Koszul homology of an explicit irregular sequence of quadratic polynomials. The corresponding Poincaré series turns out to be related to the Rogers–Ramanujan identity.
The stable Khovanov-Rozansky homology of torus knots has been conjecturally described as the Koszul homology of an explicit non-regular sequence of polynomials. We verify this conjecture against newly available computational data for sl(3)-homology. Special attention is paid to torsion. In addition, explicit conjectural formulae are given for the sl(N)-homology of (3,m)-torus knots for all N and m, which are motivated by higher categorified Jones-Wenzl projectors. Structurally similar formulae are proven for Heegard-Floer homology.
We collect a list of known four-dimensional Fano manifolds and compute their quantum periods. This list includes all four-dimensional Fano manifolds of index greater than one, all four-dimensional toric Fano manifolds, all four-dimensional products of lower-dimensional Fano manifolds, and certain complete intersections in projective bundles.