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We explicitly compute the equations and components of the jet schemes of a monomial subscheme of affine space from an algebraic perspective. [ABSTRACT FROM AUTHOR]

For projective codimension two surfaces and threefolds whose singular locus is one dimensional, we get the sharp Castelnuovo-Mumford regularity bound in terms of degrees of defining equations and give the classification of nearly extremal cases. This is a generalization of the result of Bertram et al. [ABSTRACT FROM AUTHOR]

On vérifie, en degré d ≤ 11, la conjecture suivante:“le schéma de Hilbert des courbes lisses connexes de degré d et de genre g de l'espace projectifá trois dimensions sur le corps des complexes est irréductible pour g ≤ 2d − 9” ABSTRACT We check that the following conjecture holds true for d ≤ 11 :“the Hilbert scheme of smooth and connected space curves of degree d and genus g is irreducible provided that g ≤ 2d − 9.” [ABSTRACT FROM AUTHOR]