Rank-two stable sheaves with odd determinant on Fano threefolds of genus nine

Brambilla, Maria Chiara; Faenzi, Daniele

doi:10.1007/s00209-012-1131-8

Cited by 9 publications

(18 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For X 18 , F is the moduli space of rank 3 stable vector bundles with another certain special property over a genus 2 curve. In view of Conjecture 12.4 and Theorem 12.5, the notion of the fiber of the "categorical period" map not only recovers the classical results on fibers of period maps for X 10 in [DIM12], X 14 in [MT98] and [Kuz03], and X 16 in [BF13], but also reproves the very recent result in [FV21] for X 18 .…”

Section: {Ku(x)}/ ∼supporting

confidence: 78%

Categorical Torelli theorems for Gushel--Mukai threefolds

Jacovskis¹,

Lin²,

Liu³

et al. 2021

Preprint

View full text Add to dashboard Cite

The Kuznetsov component Ku(X) of a Gushel-Mukai (GM) threefold has two numerical (−1)-classes with respect to the Euler form. We describe the Bridgeland moduli spaces for stability conditions on Kuznetsov components with respect to each of the (−1)-classes and prove refined and birational categorical Torelli theorems in terms of Ku(X). We also prove a categorical Torelli theorem for special GM threefolds. We study the smoothness and singularities on Bridgeland moduli spaces for all smooth GM threefolds and use this to prove a conjecture of Kuznetsov-Perry in dimension three under a mild assumption. Finally, we use our moduli spaces to restate a conjecture of Debarre-Iliev-Manivel regarding fibers of the period map for ordinary GM threefolds. We also prove the restatement of this conjecture infinitesimally using Hochschild (co)homology.

show abstract

Section: {Ku(x)}/ ∼supporting

confidence: 78%

Categorical Torelli theorems for Gushel--Mukai threefolds

Jacovskis¹,

Lin²,

Liu³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Here, we will focus on the case H 3 (X ) = 0 (recall that some cases with H 3 (X ) = 0 were studied in [BF08] and [BF12]). This vanishing takes place if and only if g = 12.…”

Section: Odd Instanton Bundles On Prime Fano Threefoldsmentioning

confidence: 99%

“…However in some cases, notably when X is rational, one can rely on a semiorthogonal decomposition of D b (X ) containing a subcategory equivalent to D b (Γ), where Γ is a curve whose Jacobian is isomorphic to the intermediate Jacobian of X . We investigated MI X (k) in terms of this curve already in [BF08] (for threefolds of index 1 and genus 7) and in [BF12] (for index 1 and genus 9). Here we carry out a similar analysis in the case of Del Pezzo threefolds of degree 4, see Theorem E.…”

Section: Introductionmentioning

confidence: 99%

Even and odd instanton bundles on Fano threefolds of Picard number one

Faenzi

2013

manuscripta math.

Self Cite

View full text Add to dashboard Cite

We consider an analogue of the notion of instanton bundle on the projective 3-space, consisting of a class of rank-2 vector bundles defined on smooth Fano threefolds X of Picard number one, having even or odd determinant according to the parity of K X .We first construct a well-behaved irreducible component of their moduli spaces. Then, when the intermediate Jacobian of X is trivial, we look at the associated monads, hyperdeterminants and nets of quadrics. We also study one case where the intermediate Jacobian of X is non-trivial, namely when X is the intersection of two quadrics in 5 , relating instanton bundles on X to vector bundles of higher rank on a the curve of genus 2 naturally associated 2000 Mathematics Subject Classification. Primary 14J60, 14F05. Key words and phrases. Instanton bundles, monads, Fano threefolds, semiorthogonal decomposition, nets of quadrics.The author was partially supported by ANR-09-JCJC-0097-0 INTERLOW and ANR GEOLMI.1 * 3 (−r X ) ≃ 1 , * 2 (−r X ) ≃ 2 . Set U = Hom X ( 2 , 3 ), and note that U ≃ Hom X ( 1 , 2 ).

show abstract

“…To be honest, most of the information provided by Theorem 1.1.1 can be found in the literature (see [AK77], [DR76], [FN89], [Ten74], [Isk80], [Put82], [Mar81], [Ili03], [IM07], [BF13], etc). Our goal was, in a sense, in collecting all the results together, and cleaning things a bit.…”

Section: Introductionmentioning

confidence: 99%

“…One new improvement here is the case ι(X) = 1 and g(X) = 10, where originally in [IM07, Proposition 3] a description of S(X) was known for general X only. Another improvement is the case ι(X) = 1 and g(X) = 9 where it was previously known that S(X) is a projectivization of a vector bundle over a curve of genus 3, but simplicity of the vector bundle was known only for a general threefold X (see [BF13,§3]). Also, our proof for the even genus cases, i.e.…”

mentioning

confidence: 99%

Hilbert schemes of lines and conics and automorphism groups of Fano threefolds

2018

View full text Add to dashboard Cite

We discuss various results on Hilbert schemes of lines and conics and automorphism groups of smooth Fano threefolds of Picard rank 1. Besides a general review of facts well known to experts, the paper contains some new results, for instance, we give a description of the Hilbert scheme of conics on any smooth Fano threefold of index 1 and genus 10. We also show that the action of the automorphism group of a Fano threefold X of index 2 (respectively, 1) on an irreducible component of its Hilbert scheme of lines (respectively, conics) is faithful if the anticanonical class of X is very ample except for some explicit cases.We use these faithfulness results to prove finiteness of the automorphism groups of most Fano threefolds and classify explicitly all Fano threefolds with infinite automorphism group. We also discuss a derived category point of view on the Hilbert schemes of lines and conics, and use it to identify some of them.

show abstract

Rank-two stable sheaves with odd determinant on Fano threefolds of genus nine

Cited by 9 publications

References 26 publications

Categorical Torelli theorems for Gushel--Mukai threefolds

Categorical Torelli theorems for Gushel--Mukai threefolds

Even and odd instanton bundles on Fano threefolds of Picard number one

Hilbert schemes of lines and conics and automorphism groups of Fano threefolds

Contact Info

Product

Resources

About