Geometry of Syzygies via Poncelet Varieties

Giovanna Ilardi; Paola Supino; Jean Vallès

Bollettino dell'Unione Matematica Italiana (2009)

  • Volume: 2, Issue: 3, page 579-589
  • ISSN: 0392-4041

Abstract

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We consider the Grassmannian 𝔾 r ( k , n ) of ( k + 1 ) -dimensional linear subspaces of V n = H 0 ( 1 , 𝒪 1 ( n ) ) . We define 𝔛 k , r , d as the classifying space of the k -dimensional linear systems of degree n on 1 , whose bases realize a fixed number r of polynomial relations of fixed degree d , say r syzygies of degree d . Firstly, we compute the dimension of 𝔛 k , r , d . In the second part we make a link between 𝔛 k , r , d and the Poncelet varieties. In particular, we prove that the existence of linear syzygies implies the existence of singularities on the Poncelet varieties.

How to cite

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Ilardi, Giovanna, Supino, Paola, and Vallès, Jean. "Geometry of Syzygies via Poncelet Varieties." Bollettino dell'Unione Matematica Italiana 2.3 (2009): 579-589. <http://eudml.org/doc/290555>.

@article{Ilardi2009,
abstract = {We consider the Grassmannian $\mathbb\{G\}r(k,n)$ of $(k+1)$-dimensional linear subspaces of $V_\{n\} = H^\{0\} (\mathbb\{P\}^\{1\}, \mathcal\{O\}_\{\mathbb\{P\}_\{1\}\} (n))$. We define $\mathfrak\{X\}_\{k,r,d\}$ as the classifying space of the $k$-dimensional linear systems of degree $n$ on $\mathbb\{P\}^\{1\}$, whose bases realize a fixed number $r$ of polynomial relations of fixed degree $d$, say $r$ syzygies of degree $d$. Firstly, we compute the dimension of $\mathfrak\{X\}_\{k,r,d\}$. In the second part we make a link between $\mathfrak\{X\}_\{k,r,d\}$ and the Poncelet varieties. In particular, we prove that the existence of linear syzygies implies the existence of singularities on the Poncelet varieties.},
author = {Ilardi, Giovanna, Supino, Paola, Vallès, Jean},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {579-589},
publisher = {Unione Matematica Italiana},
title = {Geometry of Syzygies via Poncelet Varieties},
url = {http://eudml.org/doc/290555},
volume = {2},
year = {2009},
}

TY - JOUR
AU - Ilardi, Giovanna
AU - Supino, Paola
AU - Vallès, Jean
TI - Geometry of Syzygies via Poncelet Varieties
JO - Bollettino dell'Unione Matematica Italiana
DA - 2009/10//
PB - Unione Matematica Italiana
VL - 2
IS - 3
SP - 579
EP - 589
AB - We consider the Grassmannian $\mathbb{G}r(k,n)$ of $(k+1)$-dimensional linear subspaces of $V_{n} = H^{0} (\mathbb{P}^{1}, \mathcal{O}_{\mathbb{P}_{1}} (n))$. We define $\mathfrak{X}_{k,r,d}$ as the classifying space of the $k$-dimensional linear systems of degree $n$ on $\mathbb{P}^{1}$, whose bases realize a fixed number $r$ of polynomial relations of fixed degree $d$, say $r$ syzygies of degree $d$. Firstly, we compute the dimension of $\mathfrak{X}_{k,r,d}$. In the second part we make a link between $\mathfrak{X}_{k,r,d}$ and the Poncelet varieties. In particular, we prove that the existence of linear syzygies implies the existence of singularities on the Poncelet varieties.
LA - eng
UR - http://eudml.org/doc/290555
ER -

References

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  1. FULTON, W. - HARRIS, J., Representation theory, a first course, GTM129, Springer-Verlag, 1991. Zbl0744.22001MR1153249DOI10.1007/978-1-4612-0979-9
  2. HARRIS, J., Algebraic Geometry, a first course, GTM133, Springer-Verlag, 1992. MR1182558DOI10.1007/978-1-4757-2189-8
  3. ILARDI, G. - SUPINO, P., Linear systems on 1 with syzygies, Comm. Algebra, 34, no. 11 (2006), 4173-4186. Zbl1109.14010MR2267579DOI10.1080/00927870600876359
  4. RAMELLA, L., La stratification du schema de Hilbert des courbes rationnelles de n par le fibré tangent restreint, C.R. Acad. Sci. Paris, 311 (1990), 181-184. Zbl0721.14014MR1065888
  5. SCHWARZENBERGER, R. L. E., Vector bundles on the projective plane, Proc. London Math. Soc., 11 (1961), 623-640. Zbl0212.26004MR137712DOI10.1112/plms/s3-11.1.623
  6. TRAUTMANN, G., Poncelet curves and associated theta characteristics, Expositiones Math., 6 (1988), 29-64. Zbl0646.14025MR927588
  7. VALLÈS, J., Fibrés de Schwarzenberger et coniques de droites sauteuses, Bull. Soc. Math. France, 128, no. 3 (2000), 433-449. MR1792477

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