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L∞ STRUCTURES ON MAPPING CONES
, 2006
"... We show that every morphism of differential graded Lie algebras induces a canonical structure of L∞algebra on its mapping cone. Moreover such a structure is compatible with the deformation functors introduced in [17]. ..."
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We show that every morphism of differential graded Lie algebras induces a canonical structure of L∞algebra on its mapping cone. Moreover such a structure is compatible with the deformation functors introduced in [17].
BRILL–GORDAN LOCI, TRANSVECTANTS AND AN ANALOGUE OF THE FOULKES CONJECTURE
, 2004
"... Abstract. The hypersurfaces of degree d in the projective space Pn correspond to points of PN, where N = ( n+d) d − 1. Now assume d = 2e is even, and let X (n,d) ⊆ PN denote the subvariety of two efold hyperplanes. We exhibit an upper bound on the Castelnuovo regularity of the ideal of X (n,d), a ..."
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Abstract. The hypersurfaces of degree d in the projective space Pn correspond to points of PN, where N = ( n+d) d − 1. Now assume d = 2e is even, and let X (n,d) ⊆ PN denote the subvariety of two efold hyperplanes. We exhibit an upper bound on the Castelnuovo regularity of the ideal of X (n,d), and show that this variety is rnormal for r ≥ 2. The latter result is representationtheoretic, and says that a certain GLn+1equivariant moprhism Sr(S2e(C n+1)) − → S2(Sre(C n+1)) is surjective for r ≥ 2; a statement which is reminiscent of the FoulkesHowe conjecture. For its proof, we reduce the statement to the case n = 1, and then show that certain transvectants of binary forms are nonzero. The latter part uses explicit calculations with Feynman diagrams and hypergeometric series. For ternary quartics and binary dics, we give explicit generators for the defining ideal of X (n,d) expressed in the language of classical invariant theory. AMS subject classification (2000): 05A15, 14F17, 20G05, 81T18.
L∞ALGEBRAS AND DEFORMATIONS OF HOLOMORPHIC MAPS
, 705
"... Abstract. We construct the deformation functor associated with a pair of morphisms of differential graded Lie algebras, and use it to study infinitesimal deformations of holomorphic maps of compact complex manifolds. In particular, using L ∞ structures, we give an explicit description of the differe ..."
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Abstract. We construct the deformation functor associated with a pair of morphisms of differential graded Lie algebras, and use it to study infinitesimal deformations of holomorphic maps of compact complex manifolds. In particular, using L ∞ structures, we give an explicit description of the differential graded Lie algebra that controls this problem. 1.
A REGULARITY RESULT FOR A LOCUS OF BRILL TYPE
, 2004
"... Abstract. Let n,d be a positive integers, with d even (say d = 2e). Write N = ( n+d) d − 1, and let X(n,d) ⊆ PN denote the locus of degree d hypersurfaces in Pn which consist of two efold hyperplanes. We calculate a bound on the Castelnuovoregularity of its defining ideal, moreover we show that ..."
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Abstract. Let n,d be a positive integers, with d even (say d = 2e). Write N = ( n+d) d − 1, and let X(n,d) ⊆ PN denote the locus of degree d hypersurfaces in Pn which consist of two efold hyperplanes. We calculate a bound on the Castelnuovoregularity of its defining ideal, moreover we show that this variety is rnormal for r ≥ 2. The latter part is proved by reducing the question to a combinatorial calculation involving Feynman diagrams and hypergeometric series. As such, it is a result of a tripartite collaboration of algebraic geometry, classical invariant theory, and modern theoretical physics.