Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1102.3618

On cubic hypersurfaces of dimension seven and eight

Atanas Iliev·Laurent Manivel
View original source ↗Read source document ↗Export metadata ↓

Abstract

Cubic sevenfolds are examples of Fano manifolds of Calabi-Yau type. We study them in relation with the Cartan cubic, the $E_6$-invariant cubic in $\PP^{26}$. We show that a generic cubic sevenfold $X$ can be described as a linear section of the Cartan cubic, in finitely many ways. To each such "Cartan representation" we associate a rank nine vector bundle on $X$ with very special cohomological properties. In particular it allows to define auto-equivalences of the non-commutative Calabi-Yau threefold associated to $X$ by Kuznetsov. Finally we show that the generic eight dimensional section of the Cartan cubic is rational.

Explore related subjects

math.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Atanas Iliev, Laurent Manivel. 2011-06-10. On cubic hypersurfaces of dimension seven and eight. https://doi.org/10.1112/plms%2Fpdt042

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Braid and Phantom

Let N be the moduli space of stable rank 2 vector bundles on a smooth projective curve of genus g>1 with fixed odd determinant. With Sebastian Torres, we previously found a semi-orthogonal decomposition of the bounded derived category of N into bounded derived categories of symmetric powers of the curve and, possibly, a phantom block. In this work, we employ the theory of weaving patterns to eliminate the possibility of a phantom, completing the proof of the decomposition conjectured by Narasimhan and, independently, by Belmans, Galkin, and Mukhopadhyay.

math.AG↗
arXiv2026

On the Alexander polynomials of conic-line arrangements

In the present paper we compute Alexander polynomials for certain classes of conic-line arrangements in the complex projective plane which are related to pencils. We prove two general results for curve arrangements coming from Halphen pencils of index $k\geq 2$. Then we apply them to the Hesse arrangement of conics and to some of its degenerations. The results are completed by computations using computer algebra. In particular, we construct conic-line arrangements which are non-reduced pencil-type arrangements and have as roots of their Alexander polynomials roots of unity of order 7. Such roots are not known and are conjectured not to exist in the class of line arrangements.

math.AG↗
arXiv2026

On the Tensor Property of Bernstein-Sato Polynomial

We prove the multiplicative Thom-Sebastiani rule for Bernstein-Sato polynomials, answering the longstanding questions of Budur and Popa. We generalize the result to the tensor of two effective divisors on the product of two arbitrary non-singular complex varieties. This also leads to a multiplicative property related to Igusa's strong monodromy conjecture. Moreover, we propose an extension of our result to Bernstein-Sato polynomials for ideals and prove it for monomial ideals.

math.AG↗