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Low pole order frames on vertical jets of the universal hypersurface

Joël Merker (2009)

Annales de l’institut Fourier

For low order jets, it is known how to construct meromorphic frames on the space of the so-called vertical k -jets J vert k ( 𝒳 ) of the universal hypersurface 𝒳 n + 1 × ( n + 1 + d ) ! ( ( n + 1 ) ! d ! ) - 1 parametrizing all projective hypersurfaces X n + 1 ( ) of degree d . In 2004, for k = n , Siu announced that there exist two constants c n 1 and c n 1 such that the twisted tangent bundle T J vert n ( 𝒳 ) 𝒪 n + 1 ( c n ) 𝒪 ( n + 1 + d ) ! ( ( n + 1 ) ! d ! ) - 1 ( c n ) is generated at every point by its global sections. In the present article, we establish this property outside a certain exceptional algebraic subset Σ J vert n ( 𝒳 ) defined by the vanishing of certain Wronskians,...

Monotone Hurwitz Numbers and the HCIZ Integral

I. P. Goulden, Mathieu Guay-Paquet, Jonathan Novak (2014)

Annales mathématiques Blaise Pascal

In this article, we prove that the complex convergence of the HCIZ free energy is equivalent to the non-vanishing of the HCIZ integral in a neighbourhood of z = 0 . Our approach is based on a combinatorial model for the Maclaurin coefficients of the HCIZ integral together with classical complex-analytic techniques.

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