Yukari Ito

McKay correspondence and Hilbert schemes in dimension three

Abstract Let G be a nontrivial finite subgroup of SL n ( C ) . Suppose that the quotient singularity C n /G has a crepant resolution π: X→ C n /G ( i.e. K X = O X ) . There is a slightly imprecise conjecture, called the McKay correspondence, stating that there is a relation between the Grothendieck group (or (co)homology group) of X and the representations (or conjugacy classes) of G with a “certain compatibility” between the intersection product and the tensor product (see e.g. [22]). The purpose of this paper is to give more precise formulation of the conjecture when X can be given as a certain variety associated with the Hilbert scheme of points in C n . We give the proof of this new conjecture for an abelian subgroup G of SL 3 ( C ) .

A case report of primary fibrosarcoma of the liver

SummaryA 75-year-old woman was admitted to our hospital complaining of right hypochondrial pain. Echo sonography and computed tomography demonstrated a large tumor with irregular internal density in the right lobe of the liver. Angiography revealed a moderately hypervascular tumor. She was treated with transcatheter arterial embolization. Three weeks later, the tumor ruptured. She died of accompanying acute myocardial infarction seven months after the onset of the illness. Autopsy revealed primary fibrosarcoma of the liver. The tumor appearance varied from firm whitish to soft myxomatous. A part of the tumor showed hemorrhagic necrosis. There was no intrahepatic metastasis. The tumor tissue was composed of spindle shaped cells and immunohistochemically stained with vimentin.

THE MCKAY CORRESPONDENCE – A BRIDGE FROM ALGEBRA TO GEOMETRY

AbstractIn this paper, we discuss the McKay correspondence as a relation between geometry of a resolution of quotient singularity and algebra of the group which acts. To see an example of the generalized McKay correspondence, we consider concrete constructions of crepant resolutions for any finite subgroup of SL(3, ℂ). Moreover, we draw an atlas of the generalized McKay correspondence to look around inside this world.

Multigraded linear series and recollement

Given a scheme Y equipped with a collection of globally generated vector bundles

G/N

E_1, \ldots , E_n

N

E1,…,En, we study the universal morphism from Y to a fine moduli space

McKay correspondence and Hilbert schemes

{\mathcal {M}}(E)