## Restrictions of Steiner bundles and divisors on the Hilbert scheme of points in the plane (2012)

Citations: | 1 - 1 self |

### BibTeX

@MISC{Huizenga12restrictionsof,

author = {Jack W. Huizenga},

title = {Restrictions of Steiner bundles and divisors on the Hilbert scheme of points in the plane},

year = {2012}

}

### OpenURL

### Abstract

### Citations

265 | Geometry of algebraic curves - Arbarello, Cornalba, et al. - 1985 |

169 |
Spindler,H.: Vector Bundles on Complex Projective Space
- Okonek, Schneider
- 1980
(Show Context)
Citation Context ...resting discussion of many of their properties. Recall that the slope μ(E) of a vector bundle E is given by c1(E)/ rk E, and that E is called semistable if every subbundle F ⊂ E has μ(F ) ≤ μ(E). See =-=[15]-=- for other basic facts about vector bundles that we will use throughout the paper. Observe that the slope of the bundle E given by the above resolution is μ(E) = s/r. The next result classifies the sl... |

103 |
Algebraic families on an algebraic surface
- Fogarty
- 1968
(Show Context)
Citation Context ...e we recall some basic facts about divisor and curve classes on the Hilbert scheme P 2[n] of n points in the plane. Details for the results in this section can be found in [13, Chapter 3]. By Fogarty =-=[10]-=-, P 2[n] is a smooth projective variety of dimension 2n. Since P 2 has irregularity q = 0, Fogarty [11] shows that Pic P 2[n] = ZH ⊕ Z(Δ/2), where H is the locus of schemes meeting a fixed line and Δ ... |

83 | Stable G-bundles and projective connections - Faltings - 1993 |

50 | Algebraic geometry, volume 52 of Graduate Texts in Mathematics - Hartshorne - 1977 |

34 |
Potier: Fibrés stables et fibrés exceptionnels sur P2
- Drezet, Le
- 1985
(Show Context)
Citation Context ...inimal slope among vector bundles with χ(E) = n ∙ rk E. If such a bundle has interpolation, it may yield a divisor spanning the edge of the effective cone. 26Drezet and Le Potier’s classification in =-=[5]-=- of the possible numerical invariants of stable vector bundles allows one to determine the minimal slope in the preceding construction, and this slope agrees with the conjectured slope in the Steiner ... |

33 |
Cayley-Bacharach Theorems and Conjectures
- Eisenbud, Green, et al.
- 1996
(Show Context)
Citation Context ...) is irreducible, (2) dominates P 2[n] , and (3) dominates P 2[m] . Let us first show that the proposition implies the theorem. We recall two facts for use in the proof. Theorem 9.3 (Cayley-Bacharach =-=[7]-=-). Let C1, C2 ⊂ P 2 be plane curves of degrees d, e, and suppose that the intersection Γ = C1 ∩ C2 is zero-dimensional. Let Γ ′ and Γ ′′ be subschemes of Γ residual to one another in Γ, and set s = d ... |

27 |
Semistable sheaves on projective varieties and their restriction to curves
- Mehta, Ramanathan
- 1982
(Show Context)
Citation Context ...complete intersection curve of sufficiently high degree, then it is known that E|C will be semistable; various results to this effect have been given by several authors including Mehta and Ramanathan =-=[14]-=- and Flenner [9]. The general theory does not provide good bounds on how large the degree of C must be, however; furthermore, it also does not usually address what happens for specific types of curves... |

19 |
Restrictions of semistable bundles on projective varieties
- FLENNER
- 1984
(Show Context)
Citation Context ...tion curve of sufficiently high degree, then it is known that E|C will be semistable; various results to this effect have been given by several authors including Mehta and Ramanathan [14] and Flenner =-=[9]-=-. The general theory does not provide good bounds on how large the degree of C must be, however; furthermore, it also does not usually address what happens for specific types of curves, for instance r... |

8 |
Algebraic families on an algebraic surface II: Picard scheme of the punctual Hilbert scheme
- Fogarty
- 1974
(Show Context)
Citation Context ...n the plane. Details for the results in this section can be found in [13, Chapter 3]. By Fogarty [10], P 2[n] is a smooth projective variety of dimension 2n. Since P 2 has irregularity q = 0, Fogarty =-=[11]-=- shows that Pic P 2[n] = ZH ⊕ Z(Δ/2), where H is the locus of schemes meeting a fixed line and Δ is the locus of nonreduced schemes. Dually, consider the following curves on P 2[n] , each parameterize... |

7 |
The geometry of syzygies, volume 229 of Graduate Texts in Mathematics
- Eisenbud
- 2005
(Show Context)
Citation Context ...ional complete intersection) everything works nicely. The other result we will need describes the minimal resolution of the ideal sheaf of a general collection of n points in P 2 . Theorem 9.4 (Gaeta =-=[6]-=-). If n = r(r + 1)/2 + s with 0 ≤ s ≤ r, then the ideal sheaf IΓ of a general Γ ∈ P 2[n] admits a resolution or 0 → O P 2(−r − 1) r−2s ⊕ O P 2(−r − 2) s → O P 2(−r) r−s+1 → IΓ → 0 0 → O P 2(−r − 2) s ... |

4 | Projective duality and homogeneous spaces, volume 133 of Encyclopaedia of Mathematical Sciences - Tevelev - 2005 |

2 | Cokernel bundles and Fibonacci bundles
- Brambilla
(Show Context)
Citation Context ... E to be locally free, it is necessary and sufficient that either s = 0 or r ≥ N. These are some of the simplest vector bundles on P N , and much is known about them; we refer the reader to Brambilla =-=[3]-=- for an interesting discussion of many of their properties. Recall that the slope μ(E) of a vector bundle E is given by c1(E)/ rk E, and that E is called semistable if every subbundle F ⊂ E has μ(F ) ... |

2 | The stability of certain vector bundles on P n in “Complex Algebraic Varieties - Bohnhorst, Spindler - 1992 |

2 | dream spaces and - Mori - 2000 |

1 |
The minimial model program for the Hilbert scheme of points on P 2 and Bridgeland stability
- Arcara, Bertram, et al.
(Show Context)
Citation Context ...e singularities in the symmetric product. In this paper, we focus on understanding the cones of effective divisors and moving curves on X [n] in the particular case X = P2 C . Our results are used in =-=[1]-=- to help describe the various birational models of P2[n] . The Picard group of P2[n] has rank 2, and is generated over Z by classes H and Δ/2, where H is the locus of subschemes meeting a given line a... |

1 | Simplicity of generic Steiner bundles - Brambilla |