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Séminaire Combinatoire et Théorie des Nombres ICJ
# On the combinatorial and algebraic structure of the arc space of a fat point

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Europe/Paris

Salle Fokko du Cloux (ICJ, Université Lyon 1)
### Salle Fokko du Cloux

#### ICJ, Université Lyon 1

Description

Fat point is the scheme defined by an ideal whose solution set is a single point (but the ideal is not necessarily maximal, so it may have multiplicity). For an algebraic variety, the arc scheme can be thought of as the scheme of all possible formal trajectories on the variety (in other words, power series solutions of the corresponding equations). This scheme is defined by an ideal in an infinite dimensional polynomial ring obtained by the original equations by formal differentiation.

The original multiplicity structure of a fat point "propagates" to its arc scheme in a nontrivial and intriguing way. For example, it is capable of encoding non trivial partition identities.

In the talk I will describe some recent results about these arc schemes including the Poincare-type series for their multiplicities for the fat points on the line and some high-dimensional fat points and the Wronskian-based description of the dual space in the case of the square of the maximal ideal.

The talk is based on joint works with Rida Ait El Manssour.