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1. CJM 2012 (vol 65 pp. 905)
Explicit Models for Threefolds Fibred by K3 Surfaces of Degree Two We consider threefolds that admit a fibration by K3 surfaces over a nonsingular curve, equipped with a divisorial sheaf that defines a polarisation of degree two on the general fibre. Under certain assumptions on the threefold we show that its relative log canonical model exists and can be explicitly reconstructed from a small set of data determined by the original fibration. Finally we prove a converse to the above statement: under certain assumptions, any such set of data determines a threefold that arises as the relative log canonical model of a threefold admitting a fibration by K3 surfaces of degree two.
Keywords:threefold, fibration, K3 surface Categories:14J30, 14D06, 14E30, 14J28 |
2. CJM 2010 (vol 62 pp. 1293)
Canonical Toric Fano Threefolds
An inductive approach to classifying all toric Fano varieties is
given. As an application of this technique, we present a
classification of the toric Fano threefolds with at worst canonical
singularities. Up to isomorphism, there are $674,\!688$ such
varieties.
Keywords:toric, Fano, threefold, canonical singularities, convex polytopes Categories:14J30, 14J30, 14M25, 52B20 |
3. CJM 2010 (vol 62 pp. 1201)
Criteria for Very Ampleness of Rank Two Vector Bundles over Ruled Surfaces
Very ampleness criteria for rank $2$ vector bundles over smooth, ruled
surfaces over rational and elliptic curves are given. The criteria are then
used to settle open existence questions for some special threefolds of low
degree.
Keywords:vector bundles, very ampleness, ruled surfaces Categories:14E05, 14J30 |
4. CJM 2004 (vol 56 pp. 495)
Coinvariant Algebras of Finite Subgroups of $\SL(3,C)$ For most of the finite subgroups of $\SL(3,\mathbf{C})$, we give explicit formulae for
the Molien series of the coinvariant algebras, generalizing McKay's formulae
\cite{M99} for subgroups of $\SU(2)$. We also study the $G$-orbit Hilbert scheme
$\Hilb^G(\mathbf{C}^3)$ for any finite subgroup $G$ of $\SO(3)$, which is known to be a
minimal (crepant) resolution of the orbit space $\mathbf{C}^3/G$. In this case the fiber
over the origin of the Hilbert-Chow morphism from $\Hilb^G(\mathbf{C}^3)$ to $\mathbf{C}^3/G$
consists of finitely many smooth rational curves, whose planar dual graph is
identified with a certain subgraph of the representation graph of $G$. This is
an $\SO(3)$ version of the McKay correspondence in the $\SU(2)$ case.
Keywords:Hilbert scheme, Invariant theory, Coinvariant algebra,, McKay quiver, McKay correspondence Categories:14J30, 14J17 |
5. CJM 2001 (vol 53 pp. 3)
The Equivariant Grothendieck Groups of the Russell-Koras Threefolds The Russell-Koras contractible threefolds are the smooth affine threefolds
having a hyperbolic $\mathbb{C}^*$-action with quotient isomorphic to the
corresponding quotient of the linear action on the tangent space at the
unique fixed point. Koras and Russell gave a concrete description of all such
threefolds and determined many interesting properties they possess.
We use this description and these properties to compute the equivariant
Grothendieck groups of these threefolds. In addition, we give certain
equivariant invariants of these rings.
Categories:14J30, 19L47 |