Deformations of hypersurfaces with non-constant Alexander polynomial

Remke Kloosterman

26-05-2022 - 14:15
Largo San Leonardo Murialdo, 1


Let X be an irreducible hypersurface in P^n of degree d. If X has isolated singularities then h^i(X)=h^i(P^n) holds for i∉{????n−1,n,2n−2}????. Most hypersurfaces with isolated singularities satisfy h^n(X)=h^n(Pn). In this talk we consider hypersurfaces with semi-weighted homogeneous (e.g., ordinary multiple points or ADE-singularities) such that h^n(X)>h^n(P^n) holds. We show that if (d,n) is not in an explicit finite list then the equianalytic deformation space of X is not T-smooth, i.e., this space is nonreduced or its dimension is larger than expected. A similar statement holds true for X if the d-fold cover Y of Pn ramified along X satisfies h^{??n+1}??(Y)>h^{??n+1}??(P^{??n+1}??). This latter result generalizes classical examples of B. Segre of degree 6m curves in P^2 with 6m^2, 7m^2, 8m^2 and 9m^2 cusps and deformation space larger than expected.

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