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| Divided powers and discriminants |
| Abstract:There exists, due to M. Haiman, an
explicite blow-up construction of the Hilbert scheme of points in the affine
complex plane. In that construction one gets the generators of an ideal $I$
in the $n$-fold symmetric product of the plane, and the blow-up of the $n$-fold
symmetric product with center definied by $I$ is the Hilbert scheme. It is interesting to see how much of this construction that can be generalized. At least one can expect a simliar construction of the Hilbert scheme of points on any smooth surface $X$. The Grothendieck-Deligne norm map $n$ is a morphism from $H$ the Hilbert scheme of points on a scheme $X$ to the symmetric product of $X$. In the talk we will discuss that morphism, and then we will construct a subscheme $Q$ of the symmetric product, whose scheme theoretic inverse by the morphism $n$ is the discriminant of the universal family $Z -> H$. If $X$ is a smooth surface then we know that the discriminant is an effective Cartier divisor, giving an induced morphism to the blow up of the symmetric product along $Q$. It is an open question wheter that induced morphism from the Hilbert scheme $H$ to the blow-up is an isomorphism. |