scholarly journals A supersingular coincidence

2021 ◽  
Author(s):  
G. K. Sankaran
Keyword(s):  
Number Theory ◽  
Moduli Space ◽  
General Type ◽  
Abelian Surfaces ◽  
Supersingular Primes

AbstractThe 15 primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71 are called the supersingular primes: they occur in several contexts in number theory and also, strikingly, they are the primes that divide the order of the Monster. It is also known that the moduli space of (1, p)-polarised abelian surfaces is of general type for these primes. In this note, we explain that apparently coincidental fact by relating it to other number-theoretic occurences of the supersingular primes.

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 The André–Oort conjecture for the moduli space of abelian surfaces

2012 ◽  
Vol 149 (2) ◽  
pp. 204-216 ◽  
Author(s):  
Jonathan Pila ◽  
Jacob Tsimerman
Keyword(s):  
Moduli Space ◽  
Abelian Surfaces

AbstractWe provide an unconditional proof of the André–Oort conjecture for the coarse moduli space 𝒜2,1 of principally polarized abelian surfaces, following the strategy outlined by Pila–Zannier.

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 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 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.

A boundary divisor in the moduli spaces of stable quintic surfaces

2017 ◽  
Vol 28 (04) ◽  
pp. 1750021 ◽  
Author(s):  
Julie Rana
Keyword(s):  
Algebraic Geometry ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Deformation Theory ◽  
General Type ◽  
Topological Invariants ◽  
Surfaces Of General Type ◽  
Wide Range ◽  
Stable Surfaces ◽  
Standing Question

We give a bound on which singularities may appear on Kollár–Shepherd-Barron–Alexeev stable surfaces for a wide range of topological invariants and use this result to describe all stable numerical quintic surfaces (KSBA-stable surfaces with [Formula: see text]) whose unique non-Du Val singularity is a Wahl singularity. We then extend the deformation theory of Horikawa to the log setting in order to describe the boundary divisor of the moduli space [Formula: see text] corresponding to these surfaces. Quintic surfaces are the simplest examples of surfaces of general type and the question of describing their moduli is a long-standing question in algebraic geometry.

scholarly journals Non-normal abelian covers

2012 ◽  
Vol 148 (4) ◽  
pp. 1051-1084 ◽  
Author(s):  
Valery Alexeev ◽  
Rita Pardini
Keyword(s):  
Abelian Group ◽  
Moduli Space ◽  
Moduli Spaces ◽  
General Type ◽  
Finite Abelian Group ◽  
Algebraic Varieties ◽  
Surfaces Of General Type ◽  
Normal Case ◽  
Abelian Cover ◽  
Abelian Covers

AbstractAn abelian cover is a finite morphism X→Y of varieties which is the quotient map for a generically faithful action of a finite abelian group G. Abelian covers with Y smooth and X normal were studied in [R. Pardini, Abelian covers of algebraic varieties, J. Reine Angew. Math. 417 (1991), 191–213; MR 1103912(92g:14012)]. Here we study the non-normal case, assuming that X and Y are S2 varieties that have at worst normal crossings outside a subset of codimension greater than or equal to two. Special attention is paid to the case of ℤr2-covers of surfaces, which is used in [V. Alexeev and R. Pardini, Explicit compactifications of moduli spaces of Campedelli and Burniat surfaces, Preprint (2009), math.AG/arXiv:0901.4431] to construct explicitly compactifications of some components of the moduli space of surfaces of general type.

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