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Search: MSC category 14J ( Surfaces and higher-dimensional varieties {For analytic theory, see 32Jxx} )

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1. CJM Online first

Zhang, Tong
Geography of Irregular Gorenstein 3-folds
In this paper, we study the explicit geography problem of irregular Gorenstein minimal 3-folds of general type. We generalize the classical Noether-Castelnuovo type inequalities for irregular surfaces to irregular 3-folds according to the Albanese dimension.

Keywords:3-fold, geography, irregular variety
Category:14J30

2. CJM Online first

Nishinou, Takeo
Toric Degenerations, Tropical Curve, and Gromov-Witten Invariants of Fano Manifolds
In this paper, we give a tropical method for computing Gromov-Witten 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, Gromov-Witten invariants, tropical curves
Category:14J45

3. 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

4. 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

5. CJM 2011 (vol 64 pp. 123)

Lee, Jae-Hyouk
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 $(r-4)_{21}$. We construct Gosset polytopes $(r-4)_{21}$ in $\operatorname{Pic} S_{r}\otimes\mathbb{Q}$ whose vertices are lines, and we identify divisor classes in $\operatorname{Pic} S_{r}$ corresponding to $(a-1)$-simplexes ($a\leq r$), $(r-1)$-simplexes and $(r-1)$-crosspolytopes of the polytope $(r-4)_{21}$. 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 $(r-4)_{21}$. 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

6. CJM 2011 (vol 63 pp. 616)

Lee, Edward
A Modular Quintic Calabi-Yau Threefold of Level 55
In this note we search the parameter space of Horrocks-Mumford quintic threefolds and locate a Calabi-Yau threefold that is modular, in the sense that the $L$-function of its middle-dimensional cohomology is associated with a classical modular form of weight 4 and level 55.

Keywords: Calabi-Yau threefold, non-rigid Calabi-Yau threefold, two-dimensional Galois representation, modular variety, Horrocks-Mumford vector bundle
Categories:14J15, 11F23, 14J32, 11G40

7. 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 Kähler 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

8. 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

9. CJM 2010 (vol 62 pp. 1201)

Alzati, Alberto; Besana, Gian Mario
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

10. 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

11. CJM 2009 (vol 61 pp. 1050)

Bertin, Marie-Amélie
Examples of Calabi--Yau 3-Folds of $\mathbb{P}^{7}$ with $\rho=1$
We give some examples of Calabi--Yau $3$-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 well-known invariants with our method. These examples are built out of Gulliksen--Neg{\aa}rd and Kustin--Miller complexes of locally free sheaves. Finally, we give two new examples of Calabi--Yau $3$-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

12. CJM 2008 (vol 60 pp. 64)

Daigle, Daniel
Classification of Linear Weighted Graphs Up to Blowing-Up and Blowing-Down
We classify linear weighted graphs up to the blowing-up and blowing-down operations which are relevant for the study of algebraic surfaces.

Keywords:weighted graph, dual graph, blowing-up, algebraic surface
Categories:14J26, 14E07, 14R05, 05C99

13. 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 blowing-ups. 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 non-vanishing of $e$ to the study of the poles of the well-known topological, Hodge and motivic zeta functions.

Categories:14E15, 14J26, 14B05, 14J17, 32S45

14. 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 base-point 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

15. CJM 2004 (vol 56 pp. 1145)

Daigle, Daniel; Russell, Peter
On Log $\mathbb Q$-Homology Planes and Weighted Projective Planes
We classify normal affine surfaces with trivial Makar-Limanov invariant and finite Picard group of the smooth locus, realizing them as open subsets of weighted projective planes. We also show that such a surface admits, up to conjugacy, one or two $G_a$-actions.

Categories:14R05, 14J26, 14R20

16. CJM 2004 (vol 56 pp. 495)

Gomi, Yasushi; Nakamura, Iku; Shinoda, Ken-ichi
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

17. 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

18. CJM 2003 (vol 55 pp. 649)

Zucconi, Francesco
Surfaces with $p_{g}=q=2$ and an Irrational Pencil
We describe the irrational pencils on surfaces of general type with $p_{g}=q=2$.

Categories:14J29, 14J25, 14D06, 14D99

19. CJM 2001 (vol 53 pp. 1309)

Steer, Brian; Wren, Andrew
The Donaldson-Hitchin-Kobayashi Correspondence for Parabolic Bundles over Orbifold Surfaces
A theorem of Donaldson on the existence of Hermitian-Einstein metrics on stable holomorphic bundles over a compact K\"ahler surface is extended to bundles which are parabolic along an effective divisor with normal crossings. Orbifold methods, together with a suitable approximation theorem, are used following an approach successful for the case of Riemann surfaces.

Categories:14J17, 57R57

20. CJM 2001 (vol 53 pp. 3)

Bell, J. P.
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

21. CJM 2000 (vol 52 pp. 1149)

Ban, Chunsheng; McEwan, Lee J.
Canonical Resolution of a Quasi-ordinary Surface Singularity
We describe the embedded resolution of an irreducible quasi-ordinary surface singularity $(V,p)$ which results from applying the canonical resolution of Bierstone-Milman to $(V,p)$. We show that this process depends solely on the characteristic pairs of $(V,p)$, as predicted by Lipman. We describe the process explicitly enough that a resolution graph for $f$ could in principle be obtained by computer using only the characteristic pairs.

Keywords:canonical resolution, quasi-ordinary singularity
Categories:14B05, 14J17, 32S05, 32S25

22. CJM 1997 (vol 49 pp. 675)

de Cataldo, Mark Andrea A.
Some adjunction-theoretic properties of codimension two non-singular subvarities of quadrics
We make precise the structure of the first two reduction morphisms associated with codimension two non-singular subvarieties of non-singular quadrics $\Q^n$, $n\geq 5$. We give a coarse classification of the same class of subvarieties when they are assumed not to be of log-general-type.}

Keywords:Adjunction Theory, classification, codimension two, conic bundles,, low codimension, non log-general-type, quadric, reduction, special, variety.
Categories:14C05, 14E05, 14E25, 14E30, 14E35, 14J10

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