Segre Cubic Research Articles

A four-dimensional cousin of the Segre cubic

This note is devoted to a special Fano fourfold defined by a four-dimensional space of skew-symmetric forms in five variables. This fourfold appears to be closely related with the classical Segre cubic and its Cremona–Richmond configuration of planes. Among other exceptional properties, it is infinitesimally rigid and has Picard number six. We show how to construct it by blow-up and contraction, starting from a configuration of five planes in a four-dimensional quadric, compatibly with the symmetry group {\mathcal S}_5 . From this construction, we are able to describe the Chow ring explicitly.

Forms of the Segre Cubic

Forms of the Segre Cubic

An instance where the major and minor arc integrals meet

We apply the circle method to obtain an asymptotic formula for the number of integral points on a certain sliced cubic hypersurface related to the Segre cubic. Unusually, the major and minor arc integrals in this application are both positive and of the same order of magnitude.

Vector-valued modular forms on a three-dimensional ball

In this paper we give a structure theorem for the module of vector valued modular forms in the case of a three dimensional ball with the action of the Picard modular group Γ 3 [ − 3 ] \Gamma _3 [\sqrt {-3}] . The corresponding modular variety of dimension 3 3 is a copy of the Segre cubic.

Automorphisms of singular three-dimensional cubic hypersurfaces

We classify three-dimensional singular cubic hypersurfaces with an action of a finite group G, which are not G-rational and have no birational structure of G-Mori fiber space with the base of positive dimension. Also we prove the $$\mathfrak {A}_{5}$$ -birational superrigidity of the Segre cubic.

Tropicalization of Classical Moduli Spaces

The image of the complement of a hyperplane arrangement under a monomial map can be tropicalized combinatorially using matroid theory. We apply this to classical moduli spaces that are associated with complex reflection arrangements. Starting from modular curves, we visit the Segre cubic, the Igusa quartic, and moduli of marked del Pezzo surfaces of degrees 2 and 3. Our primary example is the Burkhardt quartic, whose tropicalization is a 3-dimensional fan in 39-dimensional space. This effectuates a synthesis of concrete and abstract approaches to tropical moduli of genus 2 curves.

VECTOR BUNDLES, DUALITIES AND CLASSICAL GEOMETRY ON A CURVE OF GENUS TWO

Let C be a curve of genus two. We denote by [Formula: see text] the moduli space of semi-stable vector bundles of rank 3 and trivial determinant over C, and by Jd the variety of line bundles of degree d on C. In particular, J1 has a canonical theta divisor Θ. The space [Formula: see text] is a double cover of ℙ8 = |3Θ| branched along a sextic hypersurface, the Coble sextic. In the dual [Formula: see text], where J1 is embedded, there is a unique cubic hypersurface singular along J1, the Coble cubic. We prove that these two hypersurfaces are dual, inducing a non-abelian Torelli result. Moreover, by looking at some special linear sections of these hypersurfaces, we can observe and reinterpret some classical results of algebraic geometry in a context of vector bundles: the duality of the Segre–Igusa quartic with the Segre cubic, the symmetric configuration of 15 lines and 15 points, the Weddle quartic surface and the Kummer surface.

Remarks on the Segre cubic

The Igusa quartic 3-fold is the moduli space of Abelian surfaces with the level two structure, and it is known that the tangent hyperplanes cut out Kummer surfaces. In this note, we show similar results for the Segre cubic.

A quintic hypersurface in P4 with 130 nodes

In fact, N,(3) = 10, realised by the Segre Cubic ([l l]), and also N,(4) = 45, realized by the Burkhardt Quartic ([4]). These remarkable threefolds are uniquely determined by having these singularities (see [12], [9], and [S]). There is a very rich geometry associated with these varieties (see e.g. [12], [2], [S]). In [7] Hirzebruch constructed a quintic with 126 nodes. In this note we present a quintic & in

RATIONALITY OF FIELDS OF INVARIANTS OF SOME FOUR-DIMENSIONAL LINEAR GROUPS, AND AN EQUIVARIANT CONSTRUCTION RELATED TO THE SEGRE CUBIC

Let be a finite primitive linear group. We prove that if contains a normal subgroup of order 32 then the quotient variety is birationally isomorphic to , where is the Segre cubic. We also prove the rationality of for a large class of such groups (in particular, solvable groups).

Small resolutions of the Segre cubic

If X is a projective threefold with only ordinary double points, it is possible to obtain a smooth “small” resolution of X by replacing every singular point not by a surface, but by a P 1. For every double point there are two ways to do so. It is an interesting fact that such a small resolution need not be projective. In this article we shall investigate the small resolutions of the Segre cubic, i.e. the unique cubic hypersurface in P 4 with 10 ordinary double points.