Related papers: Homological projective duality for the Segre cubic
We study ramified covers of the projective plane. Given a smooth projective surface S and a generic enough projection of S to the projective plane, we get a cover of the plane ramified over a plane curve. The branch curve is usually…
Orthogonal spaces are vector spaces together with a quadratic form whose associated bilinear form is non-degenerate. Over fields of characteristic two, there are many quadratic forms associated to a given bilinear form and quadratic…
Cubic fourfolds of discriminant 24 contain special codimension-two algebraic cycles of degree 6 and self-intersection 20. Such cycles may be represented by singular scrolls or del Pezzo surfaces. A discriminant 24 cubic fourfold gives rise…
Given a smooth genus two curve $C$, the moduli space SU$_C(3)$ of rank three semi-stable vector bundles on $C$ with trivial determinant is a double cover in $\mathbb{P}^8$ branched over a sextic hypersurface, whose projective dual is the…
Hypercubic groups in any dimension are defined and their conjugate classifications and representation theories are derived. Double group and spinor representation are introduced. A detailed calculation is carried out on the structures of…
Given two parallelisms of a projective space we describe a construction, called blending, that yields a (possibly new) parallelism of this space. For a projective double space $(\mathbb{P},\parallel_\ell,\parallel_r)$ over a quaternion skew…
Schubert calculus has been in the intersection of several fast developing areas of mathematics for a long time. Originally invented as the description of the cohomology of homogeneous spaces it has to be redesigned when applied to other…
We study the link between a compact hypersurface in $\P^{n+1}$ and the set of all its tangent planes. In this context, we identify $\P^{n+1}$ to the set of linear subspaces of codimension one by orthogonal complementarity. This gives rise…
We discuss the structure of heterotic/type II duality in four dimensions as a consequence of string-string duality in six dimensions. We emphasize the new features in four dimensions which go beyond the six dimensional vacuum structure and…
We study the mixed Hodge structure on the third homology group of a threefold which is the double cover of projective three-space ramified over a quartic surface with a double conic. We deal with the Torelli problem for such threefolds.
In analogy with the Poisson algebra of the quadratic forms on the symplectic plane, and the notion of duality in the projective plane introduced by Arnold in \cite{Arn}, where the concurrence of the triangle altitudes is deduced from the…
Frames in a separable quaternionic Hilbert space were introduced and studied in [17] to have more applications. In this paper, we extend the study of frames in quaternionic Hilbert spaces and introduce different types of duals of a frame in…
Let k be a commutative ring in which 2 is invertible. We prove that the Hermitian K-theory of quadric hypersurfaces over k admits fibration sequences relating it to the base ring and to Clifford algebras equipped with various duality…
We construct algebras from rhombohedral tilings of Euclidean space obtained as projections of certain cubical complexes. We show that these `Cubist algebras' satisfy strong homological properties, such as Koszulity and quasi-heredity,…
Quartic spectrahedra in 3-space form a semialgebraic set of dimension 24. This set is stratified by the location of its ten nodes. There are twenty maximal strata, identified recently by Degtyarev and Itenberg, via the global Torelli…
We use the cut and paste relation $[Y^{[2]}] = [Y][\mathbb{P}^m] + \mathbb{L}^2 [F(Y)]$ in $K_0(\text{Var}_k)$ of Galkin--Shinder for cubic hypersurfaces arising from projective geometry to characterize cubic hypersurfaces of sufficiently…
In this paper, we develop two new homological invariants called relative dominant dimension with respect to a module and relative codominant dimension with respect to a module. These are used to establish precise connections between Ringel…
We prove the Strominger--Yau--Zaslow and topological mirror symmetries for parabolic Hitchin systems of types B and C. In contrast to type A, a geometric reinterpretation of Springer duality is necessary. Furthermore, unlike Hitchin's…
For a given irreducible projective variety $X$, the closure of the set of all hyperplanes containing tangents to $X$ is the projectively dual variety $X^{\vee}$. We study the singular locus of projectively dual varieties of certain…
Our main objective is to demonstrate how homological perturbation theory (HPT) results over the last 40 years immediately or with little extra work give some of the Koszul duality results that have appeared in the last decade. Higher…