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.

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.

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.

scholarly journals Quadratic sequences of powers and Mohanty’s conjecture

2018 ◽  
Vol 14 (02) ◽  
pp. 479-507 ◽  
Author(s):  
Natalia Garcia-Fritz
Keyword(s):  
Elliptic Curves ◽  
Arithmetic Progression ◽  
General Type ◽  
Function Fields ◽  
Hyperelliptic Curves ◽  
Arithmetic Progressions ◽  
Surfaces Of General Type ◽  
Genus Zero ◽  
Curves Of Low Genus

We prove under the Bombieri–Lang conjecture for surfaces that there is an absolute bound on the length of sequences of integer squares with constant second differences, for sequences which are not formed by the squares of integers in arithmetic progression. This answers a question proposed in 2010 by Browkin and Brzezinski, and independently by Gonzalez-Jimenez and Xarles. We also show that under the Bombieri–Lang conjecture for surfaces, for every [Formula: see text] there is an absolute bound on the length of sequences formed by [Formula: see text]th powers with constant second differences. This gives a conditional result on one of Mohanty’s conjectures on arithmetic progressions in Mordell’s elliptic curves [Formula: see text]. Moreover, we obtain an unconditional result regarding infinite families of such arithmetic progressions. We also study the case of hyperelliptic curves of the form [Formula: see text]. These results are proved by unconditionally finding all curves of genus zero or one on certain surfaces of general type. Moreover, we prove the unconditional analogues of these arithmetic results for function fields by finding all the curves of low genus on these surfaces.

scholarly journals A note on a family of surfaces with $$p_g=q=2$$ and $$K^2=7$$

2021 ◽  
Author(s):  
Matteo Penegini ◽  
Roberto Pignatelli
Keyword(s):  
Open Subset ◽  
Moduli Space ◽  
General Type ◽  
Connected Components ◽  
Surfaces Of General Type ◽  
Alternative Construction ◽  
Albanese Map

AbstractWe study a family of surfaces of general type with $$p_g=q=2$$ p g = q = 2 and $$K^2=7$$ K 2 = 7 , originally constructed by C. Rito in [35]. We provide an alternative construction of these surfaces, that allows us to describe their Albanese map and the corresponding locus $$\mathcal {M}$$ M in the moduli space of surfaces of general type. In particular we prove that $$\mathcal {M}$$ M is an open subset, and it has three connected components, all of which are 2-dimensional, irreducible and generically smooth

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