Related papers: On cubic fourfolds with an inductive structure
Cubical type theory provides a constructive justification to certain aspects of homotopy type theory such as Voevodsky's univalence axiom. This makes many extensionality principles, like function and propositional extensionality, directly…
Motivated by questions occuring in the construction of certain twistor spaces the parameter space of conics tangent to a given quartic is investigated. For a given real quartic surface in complex $\PP ^3$ that has exactly 13 ordinary nodes…
In this note, we study linear determinantal representations of smooth plane cubics over finite fields. We give an explicit formula of linear determinantal representations corresponding to rational points. Using Schoof's formula, we count…
Given a path geometry on a surface $\mathcal{U}$, we construct a causal structure on a four-manifold which is the configuration space of non-incident pairs (point, path) on $\mathcal{U}$. This causal structure corresponds to a conformal…
We consider spherical quadrangulations -- spherical embeddings of multigraphs, possibly with loops, so that every face has boundary walk of length 4 -- in which all vertices have degree 3 or 4. Interpreting each degree 4 vertex as a…
We give an explicit presentation of the integral Chow ring of a stack of smooth plane cubics. We also determine some relations in the general case of hypersurfaces of any dimension and degree.
The hypersurface of Luroth quartic curves inside the projective space of plane quartics has degree 54. We give a proof of this fact along the lines outlined in a paper by Morley, published in 1919. Another proof has been given by Le Potier…
The problem of construction of the surfaces with given sets of the points with horizontal tangential planes is considered. Such considerations are of interest in the problem of computer simulations of the waved ocean surfaces.
We study real nonsingular projective cubic fourfolds up to deformation equivalence combined with projective equivalence and prove that they are classified by the conjugacy classes of involutions induced by the complex conjugation in the…
Non-dicritical codimension one foliations on projective spaces of dimension four or higher always have an invariant algebraic hypersurface. The proof relies on a strengthening of a result by Rossi on the algebraization/continuation of…
We show that the hypercube has a face-unfolding that tiles space, and that unfolding has an edge-unfolding that tiles the plane. So the hypercube is a "dimension-descending tiler." We also show that the hypercube cross unfolding made famous…
We confirm Beauville's conjecture that claims that if the p-th exterior power of the tangent bundle of a smooth projective variety contains the p-th power of an ample line bundle, then the variety is either the projective space or the…
We study in some detail the structure of the projective quadric Q' obtained by taking the quotient of the isotropic cone in a standard pseudo-Hermitian space H_{p,q} with respect to the positive real numbers R^+ and, further, by taking the…
We introduce and study suitable Poisson structures for four dimensional maps derived as lifts and specific periodic reductions of integrable lattice equations. These maps are Poisson with respect to these structures and the corresponding…
A systematic study of non-trivial cubic extensions of the four-dimensional Poincar\'e algebra is undertaken. Explicit examples are given with various techniques (Young tableau, characters etc).
We give a criterion for a projective surface to become a quotient of a fake projective plane. We also give a detailed information on the elliptic fibration of a $(2,3)$-elliptic surface that is the minimal resolution of a quotient of a fake…
An expository description of smooth cubic curves in the real or complex projective plane.
We show that the maximal number of planes in a complex smooth cubic fourfold in ${\mathbb P}^5$ is $405$, realized by the Fermat cubic only; the maximal number of real planes in a real smooth cubic fourfold is $357$, realized by the…
This is a survey paper on the space of symplectic structures on closed 4-manifolds, for the Proceedings ICCM 2004
Using a quadratic version of the Bott residue theorem, we give a quadratic refinement of the count of twisted cubic curves on hypersurfaces and complete intersections in a projective space.