Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0403300

On the Quantum Cohomology of some Fano threefolds and a conjecture of Dubrovin

Gianni Ciolli
View original source ↗Read source document ↗Export metadata ↓

Abstract

In the present paper the small Quantum Cohomology ring of some Fano threefolds which are obtained as one- or two-curve blow-ups from $P^3$ or the quadric $Q^3$ is explicitely computed. Because of systematic usage of the associativity property of quantum product only a very small and enumerative subset of Gromov-Witten invariants is needed. Then, for these threefolds the Dubrovin conjecture on the semisimplicity of Quantum Cohomology is proven by checking the computed Quantum Cohomology rings and by showing that a smooth Fano threefold $X$ with $b_3(X)=0$ admits a complete exceptional set of the appropriate length.

Explore related subjects

math.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Gianni Ciolli. 2004-03-23. On the Quantum Cohomology of some Fano threefolds and a conjecture of Dubrovin. https://arxiv.org/abs/math/0403300

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

KEEP EXPLORING

Related papers

arXiv2026

G3-Criteria and Applications

The G3-property of a subvariety was introduced by Hironaka-Matsumura, and plays an important role for deducing connectedness and extension results. Unfortunately, it's a rather elusive notion, which is not always easy to establish. Most of the existing work is concentrated on subvarieties of homogeneous varieties. The first goal of this article is to show that mobility assumptions on the subvariety, considered in works of Badescu, Chow, Debarre, Voisin, yield a certain partial positivity property, slightly stronger than G3, previously introduced by the author. Second, we apply the result to prove that, in numerous situations, the splitting of the normal bundle of a smooth two-codimensional subvariety implies that it is a complete intersection.

math.AG↗
arXiv2026

Nodal degeneration of chiral algebras I: Global structure and gluing formula

We define a natural extension of a universal factorization algebra $\mathcal{A}$ to families of stable punctured curves, by integrating over all semistable modifications. We prove that the resulting sheaf of factorization homology satisfies a natural gluing formula, by tensoring over a certain derived associative algebra $\mathfrak{Z}_{\mathcal{A}}^0$, generalizing the Verlinde formula for gluing of conformal blocks.

math.AG↗