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A Geometrical Construction for the Polynomial Invariants of some Reflection Groups

Sarti, Alessandra — 2005

Serdica Mathematical Journal

2000 Mathematics Subject Classification: Primary 20F55, 13F20; Secondary 14L30. We construct invariant polynomials for the reflection groups [3, 4, 3] and [3, 3, 5] by using some special sets of lines on the quadric P1 × P1 in P3. Then we give a simple proof of the well known fact that the ring of invariants are rationally generated in degree 2,6,8,12 and 2,12,20,30.

Counting lines on surfaces

Samuel BoissièreAlessandra Sarti — 2007

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

This paper deals with surfaces with many lines. It is well-known that a cubic contains 27 of them and that the maximal number for a quartic is 64 . In higher degree the question remains open. Here we study classical and new constructions of surfaces with high number of lines. We obtain a symmetric octic with 352 lines, and give examples of surfaces of degree d containing a sequence of d ( d - 2 ) + 4 skew lines.

Contraction of excess fibres between the McKay correspondences in dimensions two and three

Samuel BoissièreAlessandra Sarti — 2007

Annales de l’institut Fourier

The quotient singularities of dimensions two and three obtained from polyhedral groups and the corresponding binary polyhedral groups admit natural resolutions of singularities as Hilbert schemes of regular orbits whose exceptional fibres over the origin reveal similar properties. We construct a morphism between these two resolutions, contracting exactly the excess part of the exceptional fibre. This construction is motivated by the study of some pencils of K3 surfaces appearing as minimal resolutions...

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