arXiv subjects

Ingrid Bauer

Publications and source records attributed to Ingrid Bauer.

At least 19 recordsLinked to original sources

arXiv2022

On Rigid Manifolds of Kodaira Dimension 1

We discuss rigid compact complex manifolds of Kodaira dimension 1, arising as product-quotient varieties. First, we show that there is no free rigid action on the product of $(n-1)$ elliptic curves and a curve of genus at least two. Then, we describe the occurring groups, study the quotients and prove that there is always a suitable resolution of singularities preserving rigidity. Finally, we give a complete classification of the cases arising for minimal group orders.

math.AG↗

arXiv2021

Towards a Classification of Rigid Product Quotient Varieties of Kodaira Dimension 0

In this paper the authors study quotients of the product of elliptic curves by a rigid diagonal action of a finite group $G$. It is shown that only for $G = \operatorname{He(3)}, \mathbb Z_3^2$, and only for dimension $\geq 4$ such an action can be free. A complete classification of the singular quotients in dimension 3 and the smooth quotients in dimension $4$ is given. For the other finite groups a strong structure theorem for rigid quotients is proven.

math.AG↗

arXiv2019

Fermat's Cubic, Klein's Quartic and Rigid Complex Manifolds of Kodaira Dimension One

For each $n \geq 3$ the authors provide an $n$-dimensional rigid compact complex manifold of Kodaira dimension $1$. First they construct a series of singular quotients of products of $(n-1)$ Fermat curves with the Klein quartic, which are rigid. Then using toric geometry a suitable resolution of singularities is constructed and the deformation theories of the singular model and of the resolutions are compared, showing the rigidity of the resolutions.

math.AG↗

arXiv2018

$\mathfrak S_5$-equivariant syzygies for the Del Pezzo Surface of Degree 5

The Del Pezzo surface $Y$ of degree 5 is the blow up of the plane in 4 general points, embedded in $\mathbb{P}^5$ by the system of cubics passing through these points. It is the simplest example of the Buchsbaum-Eisenbud theorem on arithmetically-Gorenstein subvarieties of codimension 3 being Pfaffian. Its automorphism group is the symmetric group $\mathfrak S_5$. We give canonical explicit $\mathfrak S_5$-invariant Pfaffian equations through a $6 \times 6$ antisymmetric matrix. We give concrete geometric descriptions of the irreducible representations of $\mathfrak S_5$. Finally, we give $\mathfrak S_5$-invariant equations for the embedding of $Y$ inside $(\mathbb{P}^1)^5$, and show that they have the same Hilbert resolution as for the Del Pezzo of degree $4$.

math.AG↗

arXiv2018

Rigid but not infinitesimally rigid compact complex manifolds

In this paper the authors give an infinite series of rigid compact complex manifolds for each dimension $d \geq 2$ which are not infinitesimally rigid, hence giving a complete answer to a problem of Morrow and Kodaira stated in the famous book "Complex manifolds".

math.AG↗

arXiv2018

Del Pezzo Surfaces, Rigid Line Configurations and Hirzebruch-Kummer Coverings

We prove the equisingular rigidity of the singular Hirzebruch-Kummer coverings $X(n, \mathcal{L})$ of the projective plane branched on line configurations $\mathcal{L}$, satisfying some technical condition. In the case, $\mathcal{L}$ = the complete quadrangle, we give explicit equations of the Hirzebruch-Kummer covering $S_n$ (=the minimal desingularisation of $X(n, \mathcal{L})$) in a product of four Fermat curves of degree n. Since $S_n$ is the $(\mathbb{Z}/n)^5$ covering of the Del Pezzo surface $Y_5$ of degree 5 branched on the 10 lines, these equations are derived from explicit equations of the image of $Y_5$ in $(\mathbb{P}^1)^4$. Version2: We added a new section, describing more generally determinantal equations for all Del Pezzo surfaces of degree $9-k \leq 6$ as subvarieties of the k-fold product of the projective line.

math.AG↗

arXiv2016

On rigid compact complex surfaces and manifolds

This article investigates the subject of rigid compact complex manifolds. First of all we investigate the different notions of rigidity (local rigidity, global rigidity, infinitesimal rigidity, etale rigidity and strong rigidity) and the relations among them. Only for curves these notions coincide and the only rigid curve is the projective line. For surfaces we prove that a rigid surface which is not minimal of general type is either a Del Pezzo surface of degree >= 5 or an Inoue surface. We give examples of rigid manifolds of dimension n >= 3 and Kodaira dimensions 0, and 2 <=k <= n. Our main theorem is that the Hirzebruch Kummer coverings of exponent n >= 4 branched on a complete quadrangle are infinitesimally rigid. Moreover, we pose a number of questions.

math.AG↗

arXiv2015

Higher dimensional Lemniscates: the geometry of $r$ particles in $n$-space with logarithmic potentials

We prove some basic theorems concerning lemniscate configurations in an Euclidean space of dimension $ n \geq 3$. Lemniscates are defined as follows. Given m points $w_j $ in $\mathbb R^n$, consider the function $F(x)$ which is the product of the distances $ |x-w_j|$: the singular level sets of the function $F$ are called lemniscates. We show via complex analysis that the critical points of $F$ have Hessian of positivity at least $(n-1)$. This implies that, if $F$ is a Morse function, then $F$ has only local minima and saddle points with negativity 1. The critical points lie in the convex span of the points $|w_j| $ (these are absolute minima): but we made also the discovery that $F$ can also have other local minima, and indeed arbitrarily many. We discuss several explicit examples. We finally prove in the appendix that all critical points are isolated.

math.AG↗

arXiv2015

Geometry and arithmetic of primary Burniat surfaces

We study the geometry and arithmetic of so-called primary Burniat surfaces, a family of surfaces of general type arising as smooth bidouble covers of a del Pezzo surface of degree 6 and at the same time as \'etale quotients of certain hypersurfaces in a product of three elliptic curves. We give a new explicit description of their moduli space and determine their possible automorphism groups. We also give an explicit description of the set of curves of geometric genus 1 on each primary Burniat surface. We then describe how one can try to obtain a description of the set of rational points on a given primary Burniat surface $S$ defined over $\mathbb Q$. This involves an explicit description of the relevant twists of the \'etale covering of $S$ coming from the second construction mentioned above and methods for finding the set of rational points on a given twist.

math.AG↗

arXiv2015

Generalized Burniat type surfaces and Bagnera-de Franchis varieties

In this article we construct three new families of surfaces of general type with p_g = q = 0,K^2 = 6, and seven new families of surfaces of general type with p_g = q = 1, K^2 = 6, realizing 10 new fundamental groups. We also show that these families correspond to pairwise distinct irreducible connected components of the Gieseker moduli space of surfaces of general type. We achieve this using two different main ingredients. First we introduce a new class of surfaces, called generalized Burniat type surfaces, and we completely classify them (and the connected com- ponents of the moduli space containing them). Second, we introduce the notion of Bagnera-de Franchis varieties: these are the free quotients of an Abelian variety by a cyclic group (not consisting only of translations). For these we develop some basic results.

math.AG↗

arXiv2014

Bloch's conjecture for Generalized Burniat Type surfaces with $p_g=0$

The aim of this article is to prove Bloch's conjecture, asserting that the group of rational equivalence classes of zero cycles of degree 0 is trivial for surfaces with geometric genus zero, for regular generalized Burniat type surfaces. The technique is the method of "enough automorphisms" introduced by Inose-Mizukami in a simplified version due to the first author.

math.AG↗

arXiv2014

The fundamental group and torsion group of Beauville surfaces

We give a survey on the fundamental group of surfaces isogenous to a higher product. If the surfaces are regular, e.g. if they are Beauville surfaces, the first homology group is a finite group. We present a MAGMA script which calculates the first homology groups of regular surfaces isogenous to a product.

math.AG↗

arXiv2013

Product-Quotient Surfaces: new invariants and algorithms

In this article we suggest a new approach to the systematic, computer-aided construction and to the classification of product-quotient surfaces, introducing a new invariant, the integer gamma, which depends only on the singularities of the quotient model X=(C_1 x C_2)/G. It turns out that gamma is related to the codimension of the subspace of H^{1,1} generated by algebraic curves coming from the construction (i.e., the classes of the two fibers and the Hirzebruch-Jung strings arising from the minimal resolution of singularities of X). Profiting from this new insight we developped and implemented an algorithm which constructs all regular product-quotient surfaces with given values of gamma and geometric genus in the computer algebra program MAGMA. Being far better than the previous algorithms, we are able to construct a substantial number of new regular product-quotient surfaces of geometric genus zero. We prove that only two of these are of general type, raising the number of known families of product-quotient surfaces of general type with genus zero to 75. This gives evidence to the conjecture that there is an effective bound of the form gamma < Gamma(p_g,q).

math.AG↗

arXiv2013

Faithful actions of the absolute Galois group on connected components of moduli spaces

We give a canonical procedure associating to an algebraic number a first a hyperelliptic curve C_a, and then a triangle curve (D_a, G_a) obtained through the normal closure of an associated Belyi function. In this way we show that the absolute Galois group Gal(\bar{\Q} /\Q) acts faithfully on the set of isomorphism classes of marked triangle curves, and on the set of connected components of marked moduli spaces of surfaces isogenous to a higher product (these are the free quotients of a product C_1 x C_2 of curves of respective genera g_1, g_2 >= 2 by the action of a finite group G). We show then, using again the surfaces isogenous to a product, first that it acts faithfully on the set of connected components of moduli spaces of surfaces of general type (amending an incorrect proof in a previous ArXiv version of the paper); and then, as a consequence, we obtain that for every element σ\in \Gal(\bar{\Q} /\Q), not in the conjugacy class of complex conjugation, there exists a surface of general type X such that X and the Galois conjugate surface X^σ have nonisomorphic fundamental groups. Using polynomials with only two critical values, we can moreover exhibit infinitely many explicit examples of such a situation.

math.AG↗

arXiv2012

Bloch's conjecture for Inoue surfaces with p_g=0, K^2 =7

The aim of this article is to prove Bloch's conjecture (asserting that the group of rational equivalence classes of zero cycles of degree zero is trivial) for Inoue surfaces with p_g=0 and K^2 = 7. These surfaces can also be described as bidouble covers of the four nodal cubic, which allows to use the method of "enough automorphisms" introduced by Inose-Mizukami (in a simplified version).

math.AG↗

arXiv2012

Burniat-type surfaces and a new family of surfaces with p_g = 0, K^2 = 3

In this paper, one of a series devoted to the classification, the moduli spaces and the discovery of new surfaces of general type with geometric genus p_g= 0, we generalize a classical construction method due to Burniat (and revisited by Inoue), constructing surfaces with K^2 = 3 and a with a new fundamental group of order 16.

math.AG↗

arXiv2012

Inoue type manifolds and Inoue surfaces: a connected component of the moduli space of surfaces with K^2 = 7, p_g=0

We show that a family of minimal surfaces of general type with p_g = 0, K^2=7, constructed by Inoue in 1994, is indeed a connected component of the moduli space: indeed that any surface which is homotopically equivalent to an Inoue surface belongs to the Inoue family. The ideas used in order to show this result motivate us to give a new definition of varieties, which we propose to call Inoue-type manifolds: these are obtained as quotients \hat{X} / G, where \hat{X} is an ample divisor in a K(Γ, 1) projective manifold Z, and G is a finite group acting freely on \hat{X} . For these type of manifolds we prove a similar theorem to the above, even if weaker, that manifolds homotopically equivalent to Inoue-type manifolds are again Inoue-type manifolds.

math.AG↗

arXiv2012

Product-Quotient Surfaces: Result and Problems

The following is an extended version of a talk given at the Kinosaki Symposium on Algebraic Geometry in October 2011. The aim is to give an overview of product-quotient surfaces, the results that have been proven so far in collaboration with several different authors as well as pointing out some of the the problems that are still open.

math.AG↗