1. CJM 2016 (vol 68 pp. 784)
 Doran, Charles F.; Harder, Andrew

Toric Degenerations and Laurent Polynomials Related to Givental's LandauGinzburg Models
For an appropriate class of Fano complete intersections in toric
varieties, we prove that there is a concrete relationship between
degenerations to specific toric subvarieties and expressions
for Givental's LandauGinzburg models as Laurent polynomials.
As a result, we show that Fano varieties presented as complete
intersections in partial flag manifolds admit degenerations to
Gorenstein toric weak Fano varieties, and their Givental LandauGinzburg
models can be expressed as corresponding Laurent polynomials.
We also use this to show that all of the Laurent polynomials
obtained by Coates, Kasprzyk and Prince by the so called Przyjalkowski
method correspond to toric degenerations of the corresponding
Fano variety. We discuss applications to geometric transitions
of CalabiYau varieties.
Keywords:Fano varieties, LandauGinzburg models, CalabiYau varieties, toric varieties Categories:14M25, 14J32, 14J33, 14J45 

2. CJM 2016 (vol 69 pp. 338)
 Garbagnati, Alice

On K3 Surface Quotients of K3 or Abelian Surfaces
The aim of this paper is to prove that a K3 surface is the minimal
model of the quotient of an Abelian surface by a group $G$ (respectively
of a K3 surface by an Abelian group $G$) if and only if a certain
lattice is primitively embedded in its NÃ©ronSeveri group.
This allows one to describe the coarse moduli space of the K3
surfaces which are (rationally) $G$covered by Abelian or K3
surfaces (in the latter case $G$ is an Abelian group).
If either $G$ has order 2 or $G$ is cyclic and acts on an Abelian
surface, this result was already known, so we extend it to the
other cases.
Moreover, we prove that a K3 surface $X_G$ is the minimal model
of the quotient of an Abelian surface by a group $G$ if and only
if a certain configuration of rational curves is present on $X_G$.
Again this result was known only in some special cases, in particular
if $G$ has order 2 or 3.
Keywords:K3 surfaces, Kummer surfaces, Kummer type lattice, quotient surfaces Categories:14J28, 14J50, 14J10 

3. CJM 2015 (vol 68 pp. 67)
4. CJM 2015 (vol 67 pp. 961)
 Abuaf, Roland; Boralevi, Ada

Orthogonal Bundles and SkewHamiltonian Matrices
Using properties of skewHamiltonian matrices and classic
connectedness results, we prove that the moduli space
$M_{ort}^0(r,n)$ of stable rank $r$ orthogonal vector bundles
on $\mathbb{P}^2$, with Chern classes $(c_1,c_2)=(0,n)$, and trivial
splitting on the general line, is smooth irreducible of
dimension $(r2)n\binom{r}{2}$ for $r=n$ and $n \ge 4$, and
$r=n1$ and $n\ge 8$. We speculate that the result holds in
greater generality.
Keywords:orthogonal vector bundles, moduli spaces, skewHamiltonian matrices Categories:14J60, 15B99 

5. CJM 2015 (vol 67 pp. 1201)
 Aluffi, Paolo; Faber, Eleonore

Chern Classes of Splayed Intersections
We generalize the Chern class relation for the transversal intersection
of two nonsingular
varieties to a relation for possibly singular varieties, under
a splayedness assumption.
We show that the relation for the ChernSchwartzMacPherson classes
holds for two splayed hypersurfaces in a nonsingular variety,
and under a `strong splayedness' assumption for more
general subschemes. Moreover, the relation is shown to hold for
the ChernFulton classes
of any two splayed subschemes.
The main tool is a formula for Segre classes of splayed
subschemes. We also discuss the Chern class relation under the
assumption that one of the
varieties is a general very ample divisor.
Keywords:splayed intersection, ChernSchwartzMacPherson class, ChernFulton class, splayed blowup, Segre class Categories:14C17, 14J17 

6. CJM 2015 (vol 67 pp. 696)
 Zhang, Tong

Geography of Irregular Gorenstein 3folds
In this paper, we study the explicit geography problem of irregular Gorenstein minimal 3folds of general type. We generalize the classical NoetherCastelnuovo type inequalities for irregular surfaces to irregular 3folds according to the Albanese dimension.
Keywords:3fold, geography, irregular variety Category:14J30 

7. CJM 2014 (vol 67 pp. 667)
 Nishinou, Takeo

Toric Degenerations, Tropical Curve, and GromovWitten Invariants of Fano Manifolds
In this paper, we give a tropical method for computing GromovWitten
type invariants
of Fano manifolds of special type.
This method applies to those Fano manifolds which admit toric
degenerations
to toric Fano varieties with singularities allowing small resolutions.
Examples include (generalized) flag manifolds of type A, and
some moduli space
of rank two bundles on a genus two curve.
Keywords:Fano varieties, GromovWitten invariants, tropical curves Category:14J45 

8. CJM 2012 (vol 65 pp. 905)
 Thompson, Alan

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 

9. CJM 2012 (vol 65 pp. 195)
 Penegini, Matteo; Polizzi, Francesco

Surfaces with $p_g=q=2$, $K^2=6$, and Albanese Map of Degree $2$
We classify minimal surfaces of general type with $p_g=q=2$ and
$K^2=6$ whose Albanese map is a generically finite double cover.
We show that the corresponding moduli space is the disjoint union
of three generically smooth irreducible components
$\mathcal{M}_{Ia}$, $\mathcal{M}_{Ib}$, $\mathcal{M}_{II}$ of
dimension $4$, $4$, $3$, respectively.
Keywords:surface of general type, abelian surface, Albanese map Categories:14J29, 14J10 

10. CJM 2011 (vol 64 pp. 123)
 Lee, JaeHyouk

Gosset Polytopes in Picard Groups of del Pezzo Surfaces
In this article, we study the correspondence between the geometry of
del Pezzo surfaces $S_{r}$ and the geometry of the $r$dimensional Gosset
polytopes $(r4)_{21}$. We construct Gosset polytopes $(r4)_{21}$ in
$\operatorname{Pic} S_{r}\otimes\mathbb{Q}$ whose vertices are lines, and we identify
divisor classes in $\operatorname{Pic} S_{r}$ corresponding to $(a1)$simplexes ($a\leq
r$), $(r1)$simplexes and $(r1)$crosspolytopes of the polytope $(r4)_{21}$.
Then we explain how these classes correspond to skew $a$lines($a\leq r$),
exceptional systems, and rulings, respectively.
As an application, we work on the monoidal transform for lines to study the
local geometry of the polytope $(r4)_{21}$. And we show that the Gieser transformation
and the Bertini transformation induce a symmetry of polytopes $3_{21}$ and
$4_{21}$, respectively.
Categories:51M20, 14J26, 22E99 

11. CJM 2011 (vol 63 pp. 616)
 Lee, Edward

A Modular Quintic CalabiYau Threefold of Level 55
In this note we search the parameter space of HorrocksMumford quintic
threefolds and locate a CalabiYau threefold that is modular, in the
sense that the $L$function of its middledimensional cohomology is
associated with a classical modular form of weight 4 and level 55.
Keywords: CalabiYau threefold, nonrigid CalabiYau threefold, twodimensional Galois representation, modular variety, HorrocksMumford vector bundle Categories:14J15, 11F23, 14J32, 11G40 

12. CJM 2011 (vol 63 pp. 481)
 Baragar, Arthur

The Ample Cone for a K3 Surface
In this paper, we give several pictorial fractal
representations of the ample or KÃ¤hler cone for surfaces in a
certain class of $K3$ surfaces. The class includes surfaces
described by smooth $(2,2,2)$ forms in ${\mathbb P^1\times\mathbb P^1\times \mathbb P^1}$ defined over a
sufficiently large number field $K$ that have a line parallel to
one of the axes and have Picard number four. We relate the
Hausdorff dimension of this fractal to the asymptotic growth of
orbits of curves under the action of the surface's group of
automorphisms. We experimentally estimate the Hausdorff dimension
of the fractal to be $1.296 \pm .010$.
Keywords:Fractal, Hausdorff dimension, K3 surface, Kleinian groups, dynamics Categories:14J28, , , , 14J50, 11D41, 11D72, 11H56, 11G10, 37F35, 37D05 

13. CJM 2010 (vol 62 pp. 1293)
 Kasprzyk, Alexander M.

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 

14. CJM 2010 (vol 62 pp. 1201)
15. CJM 2010 (vol 62 pp. 1131)
 Kleppe, Jan O.

Moduli Spaces of Reflexive Sheaves of Rank 2
Let $\mathcal{F}$ be a coherent rank $2$ sheaf on a scheme $Y \subset \mathbb{P}^{n}$ of
dimension at least two and let $X \subset Y$ be the zero set of a section
$\sigma \in H^0(\mathcal{F})$. In this paper, we study the relationship between the
functor that deforms the pair $(\mathcal{F},\sigma)$ and the two functors that deform
$\mathcal{F}$ on $Y$, and $X$ in $Y$, respectively. By imposing some conditions on two
forgetful maps between the functors, we prove that the scheme structure of
\emph{e.g.,} the moduli scheme ${\rm M_Y}(P)$ of stable sheaves on a threefold $Y$
at $(\mathcal{F})$, and the scheme structure at $(X)$ of the Hilbert scheme of curves
on $Y$ become closely related. Using this relationship, we get criteria for the
dimension and smoothness of $ {\rm M_{Y}}(P)$ at $(\mathcal{F})$, without assuming $
{\textrm{Ext}^2}(\mathcal{F} ,\mathcal{F} ) = 0$. For reflexive sheaves on $Y=\mathbb{P}^{3}$ whose
deficiency module $M = H_{*}^1(\mathcal{F})$ satisfies $ {_{0}\! \textrm{Ext}^2}(M ,M ) = 0 $
(\emph{e.g.,} of diameter at most 2),
we get necessary and sufficient conditions of unobstructedness that coincide
in the diameter one case. The conditions are further equivalent to the
vanishing of certain graded Betti numbers of the free graded minimal
resolution of $H_{*}^0(\mathcal{F})$. Moreover, we show that every irreducible
component of ${\rm M}_{\mathbb{P}^{3}}(P)$ containing a reflexive sheaf of diameter
one is reduced (generically smooth) and we compute its dimension. We also
determine a good lower bound for the dimension of any component of ${\rm
M}_{\mathbb{P}^{3}}(P)$ that contains a reflexive stable sheaf with ``small''
deficiency module $M$.
Keywords:moduli space, reflexive sheaf, Hilbert scheme, space curve, Buchsbaum sheaf, unobstructedness, cup product, graded Betti numbers.xdvi Categories:14C05, qqqqq14D22, 14F05, 14J10, 14H50, 14B10, 13D02, 13D07 

16. CJM 2009 (vol 61 pp. 1050)
 Bertin, MarieAmélie

Examples of CalabiYau 3Folds of $\mathbb{P}^{7}$ with $\rho=1$
We give some examples of CalabiYau $3$folds with $\rho=1$ and
$\rho=2$, defined over $\mathbb{Q}$ and constructed as
$4$codimensional subvarieties of $\mathbb{P}^7$ via commutative
algebra methods. We explain how to deduce their Hodge diamond and
top Chern classes from computer based computations over some
finite field $\mathbb{F}_{p}$. Three of our examples (of degree
$17$ and $20$) are new. The two others (degree $15$ and $18$) are
known, and we recover their wellknown invariants with our
method. These examples are built out of GulliksenNeg{\aa}rd and
KustinMiller complexes of locally free sheaves.
Finally, we give two new examples of CalabiYau $3$folds of
$\mathbb{P}^6$ of degree $14$ and $15$ (defined over
$\mathbb{Q}$). We show that they are not deformation equivalent to
Tonoli's examples of the same degree, despite the fact that they
have the same invariants $(H^3,c_2\cdot H, c_3)$ and $\rho=1$.
Categories:14J32, 14Q15 

17. CJM 2008 (vol 60 pp. 64)
18. CJM 2007 (vol 59 pp. 1098)
 Rodrigues, B.

Ruled Exceptional Surfaces and the Poles of Motivic Zeta Functions
In this paper we study ruled surfaces which appear as an exceptional
surface in a succession of blowingups. In particular we prove
that the $e$invariant of such a ruled exceptional surface $E$ is
strictly positive whenever its intersection with the other
exceptional surfaces does not contain a fiber (of $E$). This fact
immediately enables us to resolve an open problem concerning an
intersection configuration on such a ruled exceptional surface
consisting of three nonintersecting sections. In the second part
of the paper we apply the nonvanishing of $e$ to the study of the
poles of the wellknown topological, Hodge and motivic zeta
functions.
Categories:14E15, 14J26, 14B05, 14J17, 32S45 

19. CJM 2005 (vol 57 pp. 724)
 Purnaprajna, B. P.

Some Results on Surfaces of General Type
In this article we prove some new results on projective normality, normal
presentation and higher syzygies for surfaces of general type, not
necessarily smooth, embedded by adjoint linear series. Some of the
corollaries of more general results include: results on property $N_p$
associated to $K_S \otimes B^{\otimes n}$ where $B$ is basepoint free and
ample divisor with $B\otimes K^*$ {\it nef}, results for pluricanonical
linear systems and results giving effective bounds for adjoint linear series
associated to ample bundles. Examples in the last section show that the results
are optimal.
Categories:13D02, 14C20, 14J29 

20. CJM 2004 (vol 56 pp. 1145)
21. CJM 2004 (vol 56 pp. 495)
 Gomi, Yasushi; Nakamura, Iku; Shinoda, Kenichi

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 HilbertChow 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 

22. CJM 2003 (vol 55 pp. 649)
23. CJM 2003 (vol 55 pp. 609)
 Moraru, Ruxandra

Integrable Systems Associated to a Hopf Surface
A Hopf surface is the quotient of the complex surface $\mathbb{C}^2
\setminus \{0\}$ by an infinite cyclic group of dilations of
$\mathbb{C}^2$. In this paper, we study the moduli spaces
$\mathcal{M}^n$ of stable $\SL (2,\mathbb{C})$bundles on a Hopf
surface $\mathcal{H}$, from the point of view of symplectic geometry.
An important point is that the surface $\mathcal{H}$ is an elliptic
fibration, which implies that a vector bundle on $\mathcal{H}$ can be
considered as a family of vector bundles over an elliptic curve. We
define a map $G \colon \mathcal{M}^n \rightarrow \mathbb{P}^{2n+1}$
that associates to every bundle on $\mathcal{H}$ a divisor, called the
graph of the bundle, which encodes the isomorphism class of the bundle
over each elliptic curve. We then prove that the map $G$ is an
algebraically completely integrable Hamiltonian system, with respect
to a given Poisson structure on $\mathcal{M}^n$. We also give an
explicit description of the fibres of the integrable system. This
example is interesting for several reasons; in particular, since the
Hopf surface is not K\"ahler, it is an elliptic fibration that does
not admit a section.
Categories:14J60, 14D21, 14H70, 14J27 

24. CJM 2001 (vol 53 pp. 1309)
25. CJM 2001 (vol 53 pp. 3)
 Bell, J. P.

The Equivariant Grothendieck Groups of the RussellKoras Threefolds
The RussellKoras 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 
