Sur les invariants des pinceaux de quintiques binaires

Abstract : Let B be the homogeneous coordinate ring of the Grassmannian of
pencils of binary quintics. We are interested in the invariant of the
natural action of SL_2 on B. The quotient variety Proj(B^SL_2) is a
natural applicant for the moduli space of quintic rational space
curves.

It is known that the Grassmannian of pencils of binary quantics of
degree d and the projective space of binary quantics of degree 2d-2
are birationally equivalent; this correspondence is SL_2-equivariant.
When d is 5, it suggests to compare the algebra B^SL_2 and the
invariant algebra of the octavic. This algebra was described with
meticulous care by T. Shioda in 1967.

We establish for B^SL_2 similar results to Shioda's ones: The algebra
B^SL_2 is the quotient of the polynomial algebra
R=C[x_1,x_2,x_3,x'_3,x_4,x_5,x'_5,x_6,x_7] with 9 indeterminates
(indices show indeterminates degrees) by an ideal generated by the
maximal Pfaffians of an alternate 5x5 matrix; We find (numerically)
the minimal free resolution of the R-module B^SL_2; Finally, we get
minimal generators of the algebra B^SL_2.

To succeed, we first extend T. Springer's formula (for the Poincaré
series of the invariant algebra of a binary quantic) to the
homogeneous coordinate ring of a Grassmannian.

The following point consists in the identification of a homogeneous
system of parameters. It is possible thanks to the Wronskian morphism
which leads to a characterization of the stability on the
Grassmannian. Then the order 4 and degree 2 covariants must be studied
which provides a few geometric statements.

Our techniques also allow to describe the invariant algebras of
pencils of cubics and quartics. What's more the Wronskian study leads
to new plethysm formulas.
Document type :
Theses
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https://tel.archives-ouvertes.fr/tel-00002255
Contributor : Matthias Meulien <>
Submitted on : Thursday, January 9, 2003 - 11:18:25 PM
Last modification on : Wednesday, January 23, 2019 - 2:39:26 PM
Long-term archiving on : Monday, September 6, 2010 - 11:38:44 AM

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Matthias Meulien. Sur les invariants des pinceaux de quintiques binaires. Mathématiques [math]. Université de Versailles-Saint Quentin en Yvelines, 2002. Français. ⟨tel-00002255⟩

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