Related papers: Cremona Orbits in $\mathbb{P}^4$ and Applications
For an imaginary quadratic field $k$ of class number $>1$, we prove that there are only finitely many isomorphism classes of rational indefinite quaternion division algebras $B$ such that the associated Shimura curve $M^B$ has $k$-rational…
Special generic maps are higher dimensional versions of Morse functions with exactly two singular points, characterizing spheres topologically except 4-dimensional cases and 4-dimensional standard spheres. The class of such maps also…
How should you choose a good set of (say) 48 planes in four dimensions? More generally, how do you find packings in Grassmannian spaces? In this article I give a brief introduction to the work that I have been doing on this problem in…
In Weyl's geometry the nonintegrability problem and difficulties in defining measuring standards are reconsidered. Approaches removing the nonintegrability of lengthin in the interior of atoms are given, so that atoms may serve as measuring…
A global description of the fine Simpson moduli spaces of $1$-dimensional sheaves supported on plane quartics is given: we describe the gluing of the Brill-Noether loci described by Dr\'ezet and Maican and show that the Simpson moduli space…
We are interested in the local extrinsic geometry of smooth surfaces in 4-space, and classify jets of Monge forms by projective transformations according to $\mathcal{A}^3$-types of their central projections.
For an (imaginary) hyperelliptic curve $\mathcal{H}$ of genus $g$, we determine a basis of the Riemann-Roch space $\mathcal{L}(D)$, where $D$ is a divisor with positive degree $n$, linearly equivalent to $P_1+\cdots+ P_j+(n-j)\Omega$, with…
In this paper, we study the divisor theory of the Simpson moduli space of semistable sheaves of dimension 1 on the projective plane. We prove that these spaces are all Mori dream spaces, and calculate their nef cones. We also study the…
We study the birational geometry of a Fano 4-fold X from the point of view of Mori dream spaces; more precisely, we study rational contractions of X. Here a rational contraction is a rational map f: X-->Y, where Y is normal and projective,…
In this work, we revisit the exact computation of the round sphere partition function of 3d $\mathcal{N}=4$ circular quiver Chern-Simons theories with mass and Fayet-Iliopoulos (FI) deformations. Utilizing the Fermi gas formalism, we derive…
We describe an infinite set of smooth projective threefolds that have equivalent derived categories but are not isomorphic, contrary to a conjecture of Kawamata. These arise as blow-ups of $\mathbb P^3$ at various configurations of 8…
Degeneracy loci of morphisms between vector bundles have been used in a wide variety of situations. We introduce a vast generalization of this notion, based on orbit closures of algebraic groups in their linear representations. A preferred…
Border bases are a generalization of Gr\"obner bases for zero-dimensional ideals in polynomial rings. In this article, we introduce border bases for a non-commutative ring of linear differential operators, namely the rational Weyl algebra.…
We study the ramification divisors of projections of a smooth projective variety onto a linear subspace of the same dimension. We prove that the ramification divisors vary in a maximal dimensional family for a large class of varieties.…
We study the problem of determining when the blowup $X \to \mathbb{P}^3$ along a smooth space curve $C$ is a Mori Dream Space. We obtain sufficient conditions, as well obstructions to the Mori dreamness of $X$ based on the external geometry…
This work looks at the theory of octonionic slice regular functions through the lens of differential topology. It proves a full-fledged version of the Open Mapping Theorem for octonionic slice regular functions. Moreover, it opens the path…
Here we generalize the Gromoll-Meyer construction of an exotic 7-sphere by producing geometric models of exotic 8, 10 and Kervaire spheres as quotients of sphere bundles over spheres by free isometric actions. We give a geometric…
We revisit the classical two-dimensional McKay correspondence in two respects: The first one, which is the main point of this work, is that we take into account of the multiplicative structure given by the orbifold product; second, instead…
We use the Weil bound of multiplicative character sums together with some recent results of N. Boston and R. Jones, to show that the critical orbit of quadratic polynomials over a finite field of $q$ elements is of length $O(q^{3/4})$,…
In this manuscript, by using Belyi maps and dessin d'enfants, we construct some concrete examples of Strebel differentials with four double poles on the Riemann sphere. As an application, we could give some explicit cone spherical metrics…