Plücker forms and the theta map

Brivio, Sonia; Verra, Alessandro

doi:10.1353/ajm.2012.0034

Cited by 10 publications

(9 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Brivio and Verra [BV12] showed that is generically injective for a general curve of genus , partially answering a conjecture of Beauville [Bea06, §6]. We apply the aforementioned construction to give the following sharpening of Brivio and Verra’s result.…”

Section: Introductionmentioning

confidence: 73%

See 1 more Smart Citation

Tangent cones to generalised theta divisors and generic injectivity of the theta map

Hitching¹,

Hoff²

2017

Compositio Math.

View full text Add to dashboard Cite

Let C be a Petri general curve of genus g and E a general stable vector bundle of rank r and slope g − 1 over C with h 0 (C, E) = r + 1. For g ≥ (2r + 2)(2r + 1), we show how the bundle E can be recovered from the tangent cone to the theta divisor Θ E at O C . We use this to give a constructive proof and a sharpening of Brivio and Verra's theorem that the theta map SU C (r ) |r Θ| is generically injective for large values of g .

show abstract

Section: Introductionmentioning

confidence: 73%

“…We apply this construction to give an improvement of a result of Brivio and Verra [BV12]. To describe this application, we need to recall some more objects.…”

Section: Introductionmentioning

confidence: 99%

Tangent cones to generalised theta divisors and generic injectivity of the theta map

Hitching¹,

Hoff²

2017

Compositio Math.

View full text Add to dashboard Cite

show abstract

“…There are also good descriptions of the image of θ for r = 2 g = 2, 3 [28] [34], r = 3, g = 2 [31] [29], r = 2, g = 4 [33]. Moreover, it has recently been shown in [12] that if C is general and g >> r then θ is generically injective.…”

Section: The Theta Mapmentioning

confidence: 99%

COHERENT SYSTEMS AND MODULAR SUBAVRIETIES OF $\mathcal{SU}_C(r)$

Bolognesi

Brivio

2012

Int. J. Math.

Self Cite

View full text Add to dashboard Cite

Let C be an algebraic smooth complex curve of genus g > 1. The object of this paper is the study of the birational structure of certain moduli spaces of vector bundles and of coherent systems on C and the comparison of different type of notions of stability arising in moduli theory. Notably we show that in certain cases these moduli spaces are birationally equivalent to fibrations over simple projective varieties, whose fibers are GIT quotients (P r−1 ) rg //PGL(r), where r is the rank of the considered vector bundles. This allows us to compare different definitions of (semi-)stability (slope stability, α-stability, GIT stability) for vector bundles, coherent systems and point sets, and derive relations between them. In certain cases of vector bundles of low rank when C has small genus, our construction produces families of classical modular varieties contained in the Coble hypersurfaces. a double covering of P 8 branched along a hypersurface of degree six C 6 called the Coble-Dolgachev sextic [25]. Our result is the following. Theorem 1.3. The Coble-Dolgachev sextic C 6 is birational to a fibration over P 4 whose fibers are Igusa quartics. More precisely, C 6 contains a four-dimensional family of Igusa quartics parametrized by an open subset of P 4 .We recall that an Igusa quartic is a modular quartic hypersurface in P 4 that is related to some classical GIT quotients (see e.g. [15]) and moduli spaces. Its dual variety is a cubic three-fold called the Segre cubic, that is isomorphic to the GIT quotient (P 1 ) 6 //PGL(2).If r = 2 and g = 3, then SU C (2) is embedded by θ in P 7 as a remarkable quartic hypersurface C 4 called the Coble quartic [27]. Our methods also allow us to give a quick proof of the following fact.

show abstract

“…These spaces are interesting by themselves as higher dimensional varieties but also for important related constructions: just to mention some, one can consider higher-rank Brill-Noether theory, Theta divisors and Theta functions and the moduli spaces of coherent systems. For surveys on these topics see, for example [3,6,7]; for some results by the authors see [5,8,[11][12][13][14]. When the curve is singular, these spaces are not in general complete.…”

Section: Introductionmentioning

confidence: 99%

Nodal curves and polarizations with good properties

Brivio

Favale

2021

Rev Mat Complut

Self Cite

View full text Add to dashboard Cite

In this paper we deal with polarizations on a nodal curve C with smooth components. Our aim is to study and characterize a class of polarizations, which we call “good”, for which depth one sheaves on C reflect some properties that hold for vector bundles on smooth curves. We will concentrate, in particular, on the relation between the $${{\underline{w}}}$$ w ̲ -stability of $${\mathcal {O}}_C$$ O C and the goodness of $${{\underline{w}}}$$ w ̲ . We prove that these two concepts agree when C is of compact type and we conjecture that the same should hold for all nodal curves.

show abstract

Plücker forms and the theta map

Cited by 10 publications

References 17 publications

Tangent cones to generalised theta divisors and generic injectivity of the theta map

Tangent cones to generalised theta divisors and generic injectivity of the theta map

COHERENT SYSTEMS AND MODULAR SUBAVRIETIES OF $\mathcal{SU}_C(r)$

Nodal curves and polarizations with good properties

Contact Info

Product

Resources

About