scholarly journals Kodaira dimension of moduli of special cubic fourfolds

2019 ◽  
Vol 2019 (752) ◽  
pp. 265-300 ◽  
Author(s):  
Sho Tanimoto ◽  
Anthony Várilly-Alvarado
Keyword(s):  
Complete Intersection ◽  
Cusp Form ◽  
Moduli Space ◽  
General Type ◽  
Kodaira Dimension ◽  
Countable Family ◽  
Prior Work ◽  
Low Weight ◽  
Modular Varieties ◽  
Cubic Fourfold

Abstract A special cubic fourfold is a smooth hypersurface of degree 3 and dimension 4 that contains a surface not homologous to a complete intersection. Special cubic fourfolds give rise to a countable family of Noether–Lefschetz divisors {{\mathcal{C}}_{d}} in the moduli space {{\mathcal{C}}} of smooth cubic fourfolds. These divisors are irreducible 19-dimensional varieties birational to certain orthogonal modular varieties. We use the “low-weight cusp form trick” of Gritsenko, Hulek, and Sankaran to obtain information about the Kodaira dimension of {{\mathcal{C}}_{d}} . For example, if {d=6n+2} , then we show that {{\mathcal{C}}_{d}} is of general type for {n>18} , {n\notin\{20,21,25\}} ; it has nonnegative Kodaira dimension if {n>13} and {n\neq 15} . In combination with prior work of Hassett, Lai, and Nuer, our investigation leaves only twenty values of d for which no information on the Kodaira dimension of {{\mathcal{C}}_{d}} is known. We discuss some questions pertaining to the arithmetic of K3 surfaces raised by our results.

scholarly journals Variation of singular Kähler–Einstein metrics: Positive Kodaira dimension

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Junyan Cao ◽  
Henri Guenancia ◽  
Mihai Păun
Keyword(s):  
General Type ◽  
Einstein Metrics ◽  
Kodaira Dimension ◽  
Einstein Metric ◽  
Canonical Bundle ◽  
Fiber Space ◽  
Generic Fiber ◽  
Lelong Numbers ◽  
Kähler Einstein Metrics

Abstract Given a Kähler fiber space p : X → Y {p:X\to Y} whose generic fiber is of general type, we prove that the fiberwise singular Kähler–Einstein metric induces a semipositively curved metric on the relative canonical bundle K X / Y {K_{X/Y}} of p. We also propose a conjectural generalization of this result for relative twisted Kähler–Einstein metrics. Then we show that our conjecture holds true if the Lelong numbers of the twisting current are zero. Finally, we explain the relevance of our conjecture for the study of fiberwise Song–Tian metrics (which represent the analogue of KE metrics for fiber spaces whose generic fiber has positive but not necessarily maximal Kodaira dimension).

scholarly journals Ulrich Bundles on Intersections of Two 4-Dimensional Quadrics

2019 ◽  
Author(s):  
Yonghwa Cho ◽  
Yeongrak Kim ◽  
Kyoung-Seog Lee
Keyword(s):  
Complete Intersection ◽  
Moduli Space ◽  
Main Ingredient ◽  
Categorical Methods

Abstract In this paper, we investigate the moduli space of Ulrich bundles on a smooth complete intersection of two $4$-dimensional quadrics in $\mathbb P^5$. The main ingredient is the semiorthogonal decomposition by Bondal–Orlov, combined with the categorical methods pioneered by Kuznetsov and Lahoz–Macrì–Stellari. Using these methods, we prove that any smooth intersection of two 4-dimensional quadrics in $\mathbb P^5$ carries an Ulrich bundle of rank $r$ for every $r \ge 2$. Moreover, we provide a description of the moduli space of stable Ulrich bundles.

scholarly journals KODAIRA DIMENSION OF SUBVARIETIES II

2006 ◽  
Vol 17 (05) ◽  
pp. 619-631 ◽  
Author(s):  
THOMAS PETERNELL
Keyword(s):  
Normal Bundle ◽  
General Type ◽  
Kodaira Dimension ◽  
Projective Manifold ◽  
Fundamental Groups ◽  
Weak Version ◽  
Joint Paper ◽  
Rationally Connected

This paper continues the study of non-general type subvarieties begun in a joint paper with Schneider and Sommese [14]. We prove uniruledness of a projective manifold containing a submanifold not of general type whose normal bundle has positivity properties and study moreover the rational quotient. We also relate the fundamental groups and a prove a cohomological criterion for a manifold to be rationally connected (weak version of a conjecture of Mumford).

scholarly journals NEW EXAMPLES OF CALABI–YAU 3-FOLDS AND GENUS ZERO SURFACES

2014 ◽  
Vol 16 (02) ◽  
pp. 1350010 ◽  
Author(s):  
GILBERTO BINI ◽  
FILIPPO F. FAVALE ◽  
JORGE NEVES ◽  
ROBERTO PIGNATELLI
Keyword(s):  
Fundamental Group ◽  
Moduli Space ◽  
General Type ◽  
Picard Number ◽  
Surfaces Of General Type ◽  
Geometric Genus ◽  
Genus Zero ◽  
New Family ◽  
Hodge Numbers ◽  
Projective Lines

We classify the subgroups of the automorphism group of the product of four projective lines admitting an invariant anticanonical smooth divisor on which the action is free. As a first application, we describe new examples of Calabi–Yau 3-folds with small Hodge numbers. In particular, the Picard number is 1 and the number of moduli is 5. Furthermore, the fundamental group is nontrivial. We also construct a new family of minimal surfaces of general type with geometric genus zero, K2 = 3 and fundamental group of order 16. We show that this family dominates an irreducible component of dimension 4 of the moduli space of the surfaces of general type.

scholarly journals K-MODULI OF CURVES ON A QUADRIC SURFACE AND K3 SURFACES

2021 ◽  
pp. 1-41
Author(s):  
KENNETH ASCHER ◽  
KRISTIN DEVLEMING ◽  
YUCHEN LIU
Keyword(s):  
Complete Intersection ◽  
Moduli Space ◽  
Moduli Spaces ◽  
K3 Surfaces ◽  
Moduli Of Curves ◽  
Quadric Surface ◽  
Intersection Curves

Abstract We show that the K-moduli spaces of log Fano pairs $\left(\mathbb {P}^1\times \mathbb {P}^1, cC\right)$ , where C is a $(4,4)$ curve and their wall crossings coincide with the VGIT quotients of $(2,4)$ , complete intersection curves in $\mathbb {P}^3$ . This, together with recent results by Laza and O’Grady, implies that these K-moduli spaces form a natural interpolation between the GIT moduli space of $(4,4)$ curves on $\mathbb {P}^1\times \mathbb {P}^1$ and the Baily–Borel compactification of moduli of quartic hyperelliptic K3 surfaces.

ℤ3-actions on Horikawa surfaces

2021 ◽  
pp. 2150097
Author(s):  
Vicente Lorenzo
Keyword(s):  
Moduli Space ◽  
General Type ◽  
Admissible Pair ◽  
Algebraic Surfaces ◽  
The Other ◽  
Connected Components ◽  
Surfaces Of General Type ◽  
Canonical Models ◽  
Normal Surfaces ◽  
Stable Surfaces

Minimal algebraic surfaces of general type [Formula: see text] such that [Formula: see text] are called Horikawa surfaces. In this note, [Formula: see text]-actions on Horikawa surfaces are studied. The main result states that given an admissible pair [Formula: see text] such that [Formula: see text], all the connected components of Gieseker’s moduli space [Formula: see text] contain surfaces admitting a [Formula: see text]-action. On the other hand, the examples considered allow us to produce normal stable surfaces that do not admit a [Formula: see text]-Gorenstein smoothing. This is illustrated by constructing non-smoothable normal surfaces in the KSBA-compactification [Formula: see text] of Gieseker’s moduli space [Formula: see text] for every admissible pair [Formula: see text] such that [Formula: see text]. Furthermore, the surfaces constructed belong to connected components of [Formula: see text] without canonical models.

scholarly journals Numerical criteria for divisors on to be ample

2009 ◽  
Vol 145 (5) ◽  
pp. 1227-1248 ◽  
Author(s):  
Angela Gibney
Keyword(s):  
Moduli Space ◽  
General Type ◽  
Topological Type ◽  
Rational Curves ◽  
Canonical Divisor ◽  
Content Type ◽  
One Dimensional ◽  
Stable Curves ◽  
The One ◽  
Ample Divisors

AbstractThe moduli space $\M _{g,n}$ of n-pointed stable curves of genus g is stratified by the topological type of the curves being parameterized: the closure of the locus of curves with k nodes has codimension k. The one-dimensional components of this stratification are smooth rational curves called F-curves. These are believed to determine all ample divisors. F-conjecture A divisor on $\M _{g,n}$ is ample if and only if it positively intersects theF-curves. In this paper, proving the F-conjecture on $\M _{g,n}$ is reduced to showing that certain divisors on $\M _{0,N}$ for N⩽g+n are equivalent to the sum of the canonical divisor plus an effective divisor supported on the boundary. Numerical criteria and an algorithm are given to check whether a divisor is ample. By using a computer program called the Nef Wizard, written by Daniel Krashen, one can verify the conjecture for low genus. This is done on $\M _g$ for g⩽24, more than doubling the number of cases for which the conjecture is known to hold and showing that it is true for the first genera such that $\M _g$ is known to be of general type.

scholarly journals The moduli space of bilevel-6 abelian surfaces

2002 ◽  
Vol 168 ◽  
pp. 113-125
Author(s):  
G. K. Sankaran ◽  
J. G. Spandaw
Keyword(s):  
Moduli Space ◽  
Modular Group ◽  
Symplectic Group ◽  
Differential Forms ◽  
Kodaira Dimension ◽  
Cusp Forms ◽  
Abelian Surfaces

AbstractWe show that the moduli space of abelian surfaces with polarisation of type (1,6) and a bilevel structure has positive Kodaira dimension and indeed pg ≥ 3. To do this we show that three of the Siegel cusp forms with character for the paramodular symplectic group constructed by Gritsenko and Nikulin are cusp forms without character for the modular group associated to this moduli problem. We then calculate the divisors of the corresponding differential forms, using information about the fixed loci of elements of the paramodular group previously obtained by Brasch.

scholarly journals Generalized twisted cubics on a cubic fourfold as a moduli space of stable objects

2018 ◽  
Vol 114 ◽  
pp. 85-117 ◽  
Author(s):  
Martí Lahoz ◽  
Manfred Lehn ◽  
Emanuele Macrì ◽  
Paolo Stellari
Keyword(s):  
Moduli Space ◽  
Twisted Cubics ◽  
Cubic Fourfold
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