Projective Fourspace Research Articles

Twists of the Burkhardt quartic threefold

We study twists of the Burkhardt quartic threefold over non-algebraically closed base fields of characteristic different from 2, 3, 5. We show they all admit quartic models in projective four-space. We identify a Galois-cohomological obstruction that measures if a given twist is birational to a moduli space of abelian varieties. This obstruction has implications for the rational points on these varieties. As a result, we see that all possible 3-level structures can be realized by abelian surfaces, whereas Kummer 3-level structures that group-theoretically may be admissible, may not be realizable over certain base fields. We give an example of a Burkhardt quartic over a bivariate function field whose desingularization has no rational points at all. Our methods are based on the representation theory of \({{\,\mathrm{Sp}\,}}_4(\mathbb {F}_3)\), Galois cohomology, and the classical algebraic geometry of the Burkhardt quartic.

On the Dynamical Degrees of Reflections on Cubic Fourfolds

We compute the dynamical degrees of certain compositions of reflections in points on a smooth cubic fourfold. Our interest in these computations stems from the irrationality problem for cubic fourfolds. Namely, we hope that they will provide numerical evidence for potential restrictions on tuples of dynamical degrees realisable on general cubic fourfolds which can be violated on the projective four-space.

The smooth surfaces in ℙ4 with few apparent triple points

We classify smooth complex projective surfaces in [Formula: see text] with [Formula: see text] apparent triple points, thus recovering and extending the results of Ascione [Sulle superficie immerse in un [Formula: see text], le cui trisecanti costituiscono complessi di [Formula: see text] ordine, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur.[Formula: see text]5[Formula: see text] 6 (1897) 162–169] and Severi [Intorno ai punti doppi impropri di una superficie generale dello spazio a quattro dimensioni, e a’ suoi punti tripli apparenti, Rend. Circ. Mat. Palermo 15 (1901) 33–51] for [Formula: see text], Marletta [Le superficie generali dell’ [Formula: see text] dotate di due punti tripli apparenti, Rend. Circ. Mat. Palermo 34 (1912) 179–186] for [Formula: see text], and Aure [The smooth surfaces in [Formula: see text] without apparent triple points, Duke Math. J. 57 (1988) 423–430] for [Formula: see text]. This is done thanks to a new projective character that can be introduced as a consequence of the main result of [K. Ranestad, On smooth plane curve fibrations in [Formula: see text], in Geometry of Complex Projective Varieties, Sem. Conf., Vol. 9 (Mediterranean, 1993), pp. 243–255; J. C. Sierra and A. L. Tironi, Some remarks on surfaces in [Formula: see text] containing a family of plane curves, J. Pure Appl. Algebra 209 (2007) 361–369; V. Beorchia and G. Sacchiero, Surfaces in [Formula: see text] with a family of plane curves, J. Pure Appl. Algebra 213 (2009) 1750–1755]. Going a bit further, we obtain some bounds on the Euler characteristic [Formula: see text] in terms of the degree [Formula: see text] and the sectional genus [Formula: see text] of a smooth surface in [Formula: see text]. For [Formula: see text], these bounds were first obtained in [A. B. Aure and K. Ranestad, The smooth surfaces of degree [Formula: see text] in [Formula: see text], in Complex Projective Geometry, London Mathematical Society Lecture Note Series, Vol. 179 (Cambridge University Press, Cambridge, 1992), pp. 32–46; K. Ranestad, On smooth surfaces of degree [Formula: see text] in the projective fourspace, Ph.D. thesis, Oslo (1988); S. Popescu, On smooth surfaces of degree [Formula: see text] in the projective fourspace, Dissertation, Saarbrücken (1993)]. Here we give a different argument based on liaison that works also for [Formula: see text] and that allows us to determine the triples [Formula: see text] of the smooth surfaces with [Formula: see text] apparent triple points.

Birationally isotrivial fiber spaces

We compute the dynamical degrees of certain compositions of reflections in points on a smooth cubic fourfold. Our interest in these computations stems from the irrationality problem for cubic fourfolds. Namely, we hope that they will provide numerical evidence for potential restrictions on tuples of dynamical degrees realisable on general cubic fourfolds which can be violated on the projective four-space.

Construction of rational surfaces of degree 12 in projective fourspace

The aim of this paper is to present a construction of smooth rational surfaces in projective fourspace with degree 12 12 and sectional genus 13 13 . In particular, we establish the existences of five different families of smooth rational surfaces in projective fourspace with the prescribed invariants.

Stable rank three vector bundles on [formula omitted] with Chern classes [formula omitted

The stable rank three vector bundles on projective fourspace with Chern classes (−2,4,0) whose first twists have no non-zero global sections form an open subset of the moduli space of all stable rank three vector bundles in this class. Furthermore, these bundles are precisely those associated with elliptic conic bundles in projective fourspace. We prove that the family of these bundles is irreducible, non-singular of dimension 28. As a corollary, we show that the family of elliptic conic bundles is irreducible of dimension 36.

Irregular elliptic surfaces of degree 12 in projective fourspace

AbstractSo far a few families of smooth irregular surfaces are known to exist in ℙ4 up to pullbacks by suitable finite morphisms from ℙ4 onto ℙ4 itself. In this paper we present two different constructions of irregular smooth minimal elliptic surfaces of degree 12 in ℙ4. The first is a monad construction while the other uses liaison. The family constructed via liaison includes the surfaces of the first family as a special case. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Vafa's formula and equivariant K-theory

Vafa's formula of the Euler characteristic of Calabi-Yau hypersurface in weighted projective four-space is identified with the Euler characteristic of S 1-equivariant K-theory of the Seifert fibration over it.

The non-existence of a smooth sectionally non-special surface of degree 11 and sectional genus 8 in the projective fourspace

The non-existence of a smooth sectionally non-special surface of degree 11 and sectional genus 8 in the projective fourspace

THREE-DIMENSIONAL VARIETIES WITH RATIONAL SECTIONS

In the paper we prove the following Theorem. A three-dimensional nonsingular variety over the field of complex numbers whose generic hyperplane sections are nonsingular rational surfaces is birationally equivalent either to projective three-space or to a cubic variety in projective four-space. Bibliography: 11 entries.

Finite Metabelian Groups and the Lines of a Projective Four-Space

Finite Metabelian Groups and the Lines of a Projective Four-Space