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Some rational subvarieties of moduli spaces of stable vector bundles

Sonia Brivio, Federico Fallucca, Filippo F. Favale

Abstract

Let X be a smooth complex irreducible projective variety of dimension $n \geq 2$ and $H$ be an ample line bundle on $X$. In this paper, we construct families of $μ_H$-stable vector bundles on $X$ having fixed determinant and rank $r$, which are generated by $r+1$ global sections, parametrized by Grassmanian varieties. This gives into the corresponding moduli spaces special subvarieties birational to Grassmannian.

Some rational subvarieties of moduli spaces of stable vector bundles

Abstract

Let X be a smooth complex irreducible projective variety of dimension and be an ample line bundle on . In this paper, we construct families of -stable vector bundles on having fixed determinant and rank , which are generated by global sections, parametrized by Grassmanian varieties. This gives into the corresponding moduli spaces special subvarieties birational to Grassmannian.
Paper Structure (9 sections, 19 theorems, 76 equations)

This paper contains 9 sections, 19 theorems, 76 equations.

Table of Contents

  1. Notations and preliminary results
  2. Moduli spaces of stable sheaves
  3. Globally generated vector bundles of rank $r$ with $r+1$ global sections
  4. Main construction
  5. Some examples for surfaces
  6. Surfaces of general type
  7. Surfaces with Kodaira dimension $0$
  8. Del Pezzo surfaces
  9. Elliptic surfaces

Key Result

Theorem 1

Let $(X,L,H,r)$ an admissible collection, then the moduli space ${\mathcal{M}}_H^s(r,L, \underline{c})$ is non-empty and it contains a subvariety birational to the Grassmannian variety $\mathop{\mathrm{Gr}}\nolimits(r+1,H^0(L))$.

Theorems & Definitions (43)

  • Theorem
  • Proposition 1.1
  • proof
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6
  • ...and 33 more