arXiv subjects
Publications and source records attributed to Hanine Awada.
We prove Libgober's divisibility relations for Oka and Alexander polynomials of symplectic curves in the complex projective plane. Along the way, we give new proofs of the divisibility relations for the Alexander polynomials of complex algebraic curves with respect to a generic line at infinity.
In this paper we explore the intersection of the Hassett divisor $\mathcal C_8$, parametrizing smooth cubic fourfolds $X$ containing a plane $P$ with other divisors $\mathcal C_i$. Notably we study the irreducible components of the intersections with $\mathcal{C}_{12}$ and $\mathcal{C}_{20}$. These two divisors generically parametrize respectively cubics containing a smooth cubic scroll, and a smooth Veronese surface. First, we find all the irreducible components of the two intersections, and describe the geometry of the generic elements in terms of the intersection of $P$ with the other surface. Then we consider the problem of rationality of cubics in these components, either by finding rational sections of the quadric fibration induced by projection off $P$, or by finding examples of reducible one-apparent-double-point surfaces inside $X$. Finally, via some Macaulay computations, we give explicit equations for cubics in each component.
We show that cubic fourfolds with lattice of algebraic 2-cycles of rank greater than 19 have abelian and finite dimensional (in the sense of Kimura) Chow motive. This also implies Abelianity and finite dimensionality of the motive of related hyperKahler varieties, such as the Fano variety of lines and the LLSvS 8fold. A similar remark allows us to show the Abelianity of the motive of an infinity of LSV 10folds, and of other hyperKahler 10folds associated to the twisted intermediate Jacobian fibration of cubic fourfolds with an associated K3 surface. After that, starting from certain 4-dimensional families of K3 surfaces, we construct two families of Fano varieties whose Chow motive is finite dimensional. Varieties from the first family are some quadric surface fibrations, and contain the finite dimensional transcendental motive of a K3 surface. Varieties from the second family are singular cubic fourfolds, and their motives are Schur-finite and Abelian in Voevodsky's triangulated category of motives.
The object of this note is the moduli spaces of cubic fourfolds (resp., Gushel-Mukai fourfolds) which contain some special rational surfaces. Under some hypotheses on the families of such surfaces, we develop a general method to show the unirationality of the moduli spaces of the $n$-pointed such fourfolds. We apply this to some codimension 1 loci of cubic fourfolds (resp., Gushel-Mukai fourfolds) appeared in the literature recently.
In this short note, we show that there exist one dimensional families of cubic fourfolds with Chow motive of abelian type and finite dimensional inside every Hassett divisor of special cubic fourfolds. This also implies abelianity and finite dimensionality of the motive of related Hyperk\"ahler varieties, such as the Fano variety of lines and the LLSvS 8fold.
Generalizing a recent construction of Yang and Yu, we study to what extent one can intersect Hassett's Noether-Lefschetz divisors $\mathcal{C}_d$ in the moduli space of cubic fourfolds $\mathcal{C}$. In particular, we exhibit arithmetic conditions on 20 indexes $d_1,\dots, d_{20}$ that assure that the divisors $\mathcal{C}_{d_1},\dots,\mathcal{C}_{d_{20}}$ all intersect one another. This allows us to produce examples of rational cubic fourfolds with an associated K3 surface with rank 20 N\'eron-Severi group, i.e. a singular K3 surface.
The aim of this short note is to define the \it universal cubic fourfold \rm over certain loci of their moduli space. Then, we propose two methods to prove that it is unirational over the Hassett divisors $\mathcal{C}_d$, in the range $8\leq d \leq 42$. By applying inductively this argument, we are able to show that, in the same range of values, $\mathcal{C}_{d,n}$ is unirational for all integer values of $n$. Finally, we observe that for explicit infinitely many values of $d$, the universal cubic fourfold over $\mathcal{C}_d$ can not be unirational.
Some classes of cubic fourfolds are birational to fibrations over $P^2$, where the fibers are rational surfaces. This is the case for cubics containing a plane (resp. an elliptic ruled surface), where the fibers are quadric surfaces (resp. del Pezzo sextic surfaces). It is known that the rationality of these cubic hypersurfaces is related to the rationality of these surfaces over the function field of $P^2$ and to the existence of rational (multi)sections of the fibrations. We study, in the moduli space of cubic fourfolds, the intersection of the divisor $C_{8}$ (resp. $C_{18}$) with $C_{14}$, $C_{26}$ and $C_{38}$, whose elements are known to be rational cubic fourfolds. We provide descriptions of the irreducible components of these intersections and give new explicit examples of rational cubics fibered in (quartic, quintic) del Pezzo surfaces or in quadric surfaces over $P^2$. We also investigate the existence of rational sections for these fibrations. Under some mild assumptions on the singularities of the fibers, these properties can be translated in terms of Brauer classes on certain surfaces.