Related papers: On singular Luroth quartics
We prove a formula for the geometric genus of splice-quotient singularities (in the sense of Neumann and Wahl). This formula enables us to compute the invariant from the resolution graph; in fact, it reduces the computation to that for…
We study the birational geometry of irreducible holomorphic symplectic varieties arising as varieties of lines of general cubic fourfolds containing a cubic scroll. We compute the ample and moving cones, and exhibit a birational…
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…
We study the algebraic exceptional set for surfaces (S,B) of log general type, when B has at least three irreducible components; we prove that in most cases it is finite or empty.
This note focuses on the problem of representing convex sets as projections of the cone of positive semidefinite matrices, in the particular case of sets generated by bivariate polynomials of degree four. Conditions are given for the convex…
Semitoric systems are a special class of completely integrable systems with two degrees of freedom that have been symplectically classified by Pelayo and Vu Ngoc about a decade ago in terms of five symplectic invariants. If a semitoric…
The space of $4 \times 4$ special orthogonal matrices with zeros on the diagonal decomposes into the union of $14$ irreducible surfaces whose intersections are beautifully encoded by the cuboctahedron. Using this decomposition, we exhibit a…
Paul Erd\H{o}s posed the question of whether every measurable planar set of infinite Lebesgue measure contains the four vertices of an isosceles trapezoid of unit area. In this paper, we provide an affirmative answer to this question.…
We explore maximal unitarity for nonplanar two-loop integrals with up to four massive external legs. In this framework, the amplitude is reduced to a basis of master integrals whose coefficients are extracted from maximal cuts. The…
We test the methods for computing the Picard group of a $K3$ surface in a situation of high rank. The examples chosen are resolutions of quartics in $\bP^3$ having 14 singularities of type $A_1$. Our computations show that the method of R.…
We calculate the dimension of the locus of elliptic surfaces over P^1 with a section and a given Picard number, in the corresponding moduli space.
The characteristic numbers of smooth plane quartics are computed using intersection theory on a component of the moduli space of stable maps. This completes the verification of Zeuthen's prediction of characteristic numbers of smooth plane…
Over the projective plane and at most two-step blowups of Hirzebruch surfaces, where there are strong full exceptional sequences of line bundles, we obtain foundational results about Gaeta resolutions of coherent sheaves by these line…
We show that the arithmetically Cohen--Macaulay (ACM) curves of degree 4 and genus 0 in ${\bold P}^4$ form an irreducible subset of the Hilbert scheme. Using this, we show that the singular locus of the corresponding component of the…
Quadratic points of a surface in the projective 3-space are the points which can be exceptionally well approximated by a quadric. They are also singularities of a 3-web in the elliptic part and of a line field in the hyperbolic part of the…
We study certain equivariant deformation components of minimally elliptic surface singularities under finite group actions. Interesting examples include cyclic quotients of simple elliptic singularities and finite group quotients of cusp…
We investigate ortho-integral (OI) hyperbolic surfaces with totally geodesic boundaries, defined by the property that every orthogeodesic (i.e. a geodesic arc meeting the boundary perpendicularly at both endpoints) has an integer…
We give explicit index-free formulae for all the degree six (and also degree four and two) algebraically independent local unitary invariant polynomials for finite dimensional k-partite pure and mixed quantum states. We carry out this by…
In this paper we study the invariant Carnot-Caratheodory metrics on $SU(2)\simeq S^3$, $SO(3)$ and $SL(2)$ induced by their Cartan decomposition and by the Killing form. Beside computing explicitly geodesics and conjugate loci, we compute…
In this article we find all (decomposable and indecomposable) $6\times 6$ linear determinantal representations of smooth Weierstrass cubics. As a corollary we verify the Kippenhahn conjecture for $M_6$.