Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0707.1605

Secant dimensions of low-dimensional homogeneous varieties

Karin Baur·Jan Draisma
View original source ↗Read source document ↗Export metadata ↓

Abstract

We completely describe the higher secant dimensions of all connected homogeneous projective varieties of dimension at most 3, in all possible equivariant embeddings. In particular, we calculate these dimensions for all Segre-Veronese embeddings of P^1 * P^1, P^1 * P^1 * P^1, and P^2 * P^1, as well as for the variety F of incident point-line pairs in P^2. For P^2 * P^1 and F the results are new, while the proofs for the other two varieties are more compact than existing proofs. Our main tool is the second author's tropical approach to secant dimensions.

Explore related subjects

math.AGmath.RT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Karin Baur, Jan Draisma. 2007-07-11. Secant dimensions of low-dimensional homogeneous varieties. https://arxiv.org/abs/0707.1605

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

KEEP EXPLORING

Related papers

arXiv2026

Canonical coordinates for moduli spaces of rank two irregular connections on curves

In this paper, we study a geometric counterpart of the cyclic vector which allow us to put a rank 2 meromorphic connection on a curve into a ``companion'' normal form. This allow us to naturally identify an open set of the moduli space of $\mathrm{GL}_2$-connections (with fixed generic spectral data, i.e. unramified, non resonant) with some Hilbert scheme of points on the twisted cotangent bundle of the curve. We prove that this map is symplectic, therefore providing Darboux (or canonical) coordinates on the moduli space, i.e. separation of variables. On the other hand, for $\mathrm{SL}_2$-connections, we give an explicit formula for the symplectic structure for a birational model given by Matsumoto. We finally detail the case of an elliptic curve with a divisor of degree $2$.

math.AG↗
arXiv2026

Group Actions from Algebraic Flops

This paper constructs derived autoequivalences associated to an algebraic flopping contraction X --> X_con, where X is quasi-projective with only mild singularities. These functors are constructed naturally using bimodule cones, and we prove these cones are locally two-sided tilting complexes by using the local-global properties and a key commutative diagram. The main result is that these autoequivalences combine to give an action of the fundamental group of an associated infinite hyperplane arrangement on the derived category of X. This generalises and simplifies \cite{DW3}, by finally removing reliance on subgroups, and it also lifts many other results from the complete local setting.

math.AG↗
arXiv2026

Unramified Gromov-Witten and Gopakumar-Vafa invariants

Kim, Kresch, and Oh defined unramified Gromov-Witten invariants. For threefolds, Pandharipande conjectured that these invariants are equal to the Gopakumar-Vafa (BPS) invariants for Fano and primitive Calabi-Yau curve classes. We prove the conjecture using a wall-crossing method developed here. As a result, we obtain an algebro-geometric construction of Gopakumar-Vafa invariants in these cases.

math.AG↗