Publications

File Date: October 2017
Books
Dissertation and Habilitationsschrift

Papers:
            Mathematics Education
                       Non-refereed Publications
                       Refereed Papers
                       Preprints
            Algebraic Geometry
                       Non-refereed Publications
                       Refereed Papers
                       Preprints


This is a list of my publications, with mini-abstracts.
There is also a short version without mini-abstracts.

Books

  1. Bauer, Th.:
    Analysis — Arbeitsbuch.
    Bezüge zwischen Schul- und Hochschulmathematik — sichtbar gemacht in Aufgaben mit kommentierten Lösungen. Springer Spektrum, Wiesbaden, 2012, ISBN: 978-3-8348-1914-7

    More information
  2. Roth, J., Bauer, Th., Koch, H., Prediger, S. (Hrsg.):
    Übergänge konstruktiv gestalten: Ansätze für eine zielgruppenspezifische Hochschuldidaktik.
    Springer Spektrum, Heidelberg, 2014, ISBN: 978-3-658-06726-7

    Webseite beim Springer-Verlag

Dissertation and Habilitationsschrift

  1. Th. Bauer:
    Linearsysteme auf Kummerflächen.
    Dissertation, Univ. Erlangen-Nürnberg, 1993.
  2. Th. Bauer:
    Seshadri constants on algebraic surfaces.
    Habilitationsschrift, Univ. Erlangen-Nürnberg, 1998.

Mathematics Education

Non-refereed Publications

  1. Th. Bauer:
    Schulmathematik und universitäre Mathematik -- Vernetzung durch Schnittstellenaufgaben zur Analysis.
    In: Hoppenbrock, A., Schreiber, S., Göller, R., Biehler, R., Büchler, B., Hochmuth, R. & Rück, H-G. (Eds.), Mathematik im Übergang Schule/Hochschule und im ersten Studienjahr: Extended Abstracts zur 2. khdm-Arbeitstagung (pp. 15-16), Kassel: Universitätsbibliothek, 2013.
    • Das Bewusstsein um die Bruchstellen zwischen Schulmathematik und universitärer Mathematik hat in den letzten Jahren stark zugenommen. Dabei ist insbesondere die Einsicht gewachsen, dass sich bei vielen Studierenden die notwendigen Verknüpfungen nicht von alleine aufbauen, sondern dass hierzu geeignete Schnittstellenaktivitäten im Studium erforderlich sind. Der Autor hat dazu Lerngelegenheiten in einem hochschuldidaktischen Projekt entwickelt, das die Vernetzung von Schulanalysis und universitärer Analysis durch spezielle Übungsaufgaben im Modul Analysis anstrebt.
  2. Th. Bauer:
    Übungsgelegenheiten im Mathematikstudium -- Erproben neuer Konzepte. [PDF]
    In: Schelhowe, H., Schaumburg, M., Jasper, J. (Eds.), Teaching is Touching the Future. Academic teaching within and across disciplines (pp. 139-139). Bielefeld: Webler, 2015.
    • In der Keynote, die diesem Beitrag zugrunde liegt, wurden drei Problemfelder des Mathematikstudiums herausgearbeitet und es wurde jeweils ein Ansatz vorgestellt, mit dem der Autor dem festgestellten Problem begegnet -- Übungen und Übungsaufgaben spielen dabei jeweils eine zentrale Rolle. (1) Doppelte Diskontinuität im Lehramtsstudium: »Warum soll ich als Lehramtsstudent/Lehramtsstudentin so etwas lernen?« -- Wie durch Schnittstellenaufgaben Bezüge zwischen Schul- und Hochschulmathematik hergestellt werden können. (2) Frustration bei Übungsaufgaben: »Die schaffe ich sowieso nicht.« -- Wie Online-Aufgaben als Brückenschlag zwischen Vorlesung und traditionellen Übungsaufgaben dienen können. (3) Ineffizient genutzte Präsenzzeit: »In der Übung schreibe ich die Musterlösungen mit.« -- Wie Studierende durch die Methode der Peer-Instruction in den Übungen aktiviert werden können.
  3. Th. Bauer, E. Kuennen:
    Building and measuring mathematical sophistication in pre-service mathematics teachers. [PDF]
    In: Göller, R., Biehler, R., Hochmuth, R., Rück, H.-G. (Eds.). Didactics of Mathematics in Higher Education as a Scientific Discipline -- Conference Proceedings (pp. 360-364). Kassel, Germany: Universitätsbibliothek Kassel, 2017.
    • We advocate that fostering mathematical sophistication should be a main role that advanced mathematics contents courses play in the university education of pre-service teachers.

Refereed Papers

  1. Th. Bauer, U. Partheil:
    Schnittstellenmodule in der Lehramtsausbildung im Fach Mathematik. [PDF] [Journal]
    Math. Semesterber. 56, 85-103 (2009)
    • This paper (written in german) is concerned with teacher education. It describes the development of a specific approach to teaching analysis courses for pre-service math teachers. It includes explicit examples and describes the experiences made so far.
  2. Th. Bauer:
    Schnittstellen bearbeiten in Schnittstellenaufgaben. [PDF] [Publisher link]
    In: Ch. Ableitinger, J. Kramer, S. Prediger (Hrsg.), Zur doppelten Diskontinuität in der Gymnasiallehrerbildung (pp. 39-56). Wiesbaden: Springer Spektrum, 2013.
    • In der aktuellen Diskussion zur doppelten Diskontinuität besteht weitgehend Einigkeit darüber, dass sich bei vielen Studierenden die Bezüge zwischen Schulmathematik und universitärer Mathematik nicht von ganz alleine einstellen, sondern dass hierfür gezielte Schnittstellenaktivitäten erforderlich sind. Absicht des vorliegenden Texts ist es, aufzuzeigen, wie solche Aktivitäten im Rahmen von Schnittstellenaufgaben gestaltet werden können und dies anhand von Beispielaufgaben zu konkretisieren.
  3. Th. Bauer:
    Schulmathematik und universitäre Mathematik -- Vernetzung durch inhaltliche Längsschnitte. [PDF] [Publisher link]
    In: H. Allmendinger, K. Lengnink, A. Vohns, G. Wickel (Hrsg.), Mathematik verständlich unterrichten. Perspektiven für Unterricht und Lehrerbildung (pp. 235-252). Wiesbaden: Springer Spektrum, 2013.
    • In diesem Aufsatz wird gezeigt, wie mittels Schnittstellenaufgaben inhaltliche Längsschnitte von der Sekundarstufe I über die gymnasiale Oberstufe zur Analysisvorlesung und darüber hinaus gebildet werden können. Das Ziel solcher Längsschnitte ist es, Studierenden Gelegenheiten zu geben, Verknüpfungen zwischen Schul- und Hochschulmathematik aufzubauen.
  4. Thomas Bauer, Wolfgang Gromes, Ulrich Partheil:
    Mathematik verstehen von verschiedenen Standpunkten aus -- Zugänge zum Krümmungsbegriff. [PDF]
    In: Hoppenbrock, A., Biehler, R., Hochmuth, R., Rück, H.-G. (Hrsg.), Lehren und Lernen von Mathematik in der Studieneingangsphase (pp. 483-499), Springer, 2016.
    • Es wird weithin davon ausgegangen, dass Lehramtsstudierende der Mathematik auf der fachinhaltlichen Seite ausreichend (oder gar »mehr als ausreichend«) für schulmathematische Erfordernisse gerüstet seien. An Beispielen wie dem Krümmungsbegriff lässt sich jedoch erkennen, dass diese Annahme nicht uneingeschränkt richtig ist: Wenn der zu einem Konzept als fachlich adäquat angesehene Standpunkt über dem im Lehramtscurriculum Erreichbaren liegt, dann kommen Lehramtsstudierende mit diesem Gegenstand in der Regel überhaupt nicht in Berührung und sind daher hierfür fachlich nicht vorbereitet. Wir betonen in diesem Text die Notwendigkeit, in solchen Situationen Zugänge auf elementaren Stufen zu finden. Dies konkretisieren wir am Beispiel des Krümmungsbegriffs und zeigen die Fruchtbarkeit der vorgestellten Zugänge für Schnittstellenaktivitäten.
  5. Th. Bauer:
    Schulmathematik und Hochschulmathematik -- was leistet der höhere Standpunkt?. [PDF]
    Der Mathematikunterricht 63, 36-45 (2017)
    • Das Mathematikstudium scheint in seinen Fachinhalten und den geforderten Denkweisen in beträchtlichem Abstand zur Schulmathematik zu liegen. Wo liegt die Relevanz einer solchen Ausbildung für angehende Lehrer und Lehrerinnen? Wo liegt ihr Nutzen? Der Beitrag geht dieser Frage in zwei Richtungen nach:   1) Wo wird vertieftes Fachwissen beim Unterrichten benötigt?   2) Was ist gemeint, wenn »mathematisches Denken« als ein Ziel der universitären Ausbildung genannt wird?

Preprints

  1. Th. Bauer, E. Müller-Hill, R. Weber:
    Fostering subject-driven professional competence of pre-service mathematics teachers -- a course conception and first results.
    Preprint 2017 (submitted)
    • We present ProfiWerk, a professionalization course geared towards pre- service Gymnasium teachers in mathematics, which is part of the preparation for an extended school-internship phase. Since the transition from university education to school practice can come with adverse discontinuity effects -- rendering, at worst, university education ineffective -- special focus is put on establishing stable connections between both mathematics content knowledge and mathematics education knowledge to the professional demands on mathematics teachers.

Algebraic Geometry

Non-refereed Publications

  1. Th. Bauer:
    Zariski chambers and stable base loci. [PDF]
    In: Y. Tschinkel (ed.), Mathematisches Institut, Georg-August-Universität Göttingen: Seminars Summer Term 2004 (pp. 75-82). Universitätsdrucke Göttingen, 2004.
    • This expository paper reports on the work with Küronya and Szemberg on the big cone of algebraic surfaces, focusing on explaining the cone decomposition.

Refereed Papers

  1. Th. Bauer:
    Projective images of Kummer surfaces. [PDF] [Journal]
    Math. Ann. 299, 155-170 (1994)
    • This paper provides projective embeddings for the Kummer surfaces associated with abelian surfaces of arbitrary polarization. The classically known special case is where the abelian surface is principally polarized.
  2. Th. Bauer, W. Barth:
    Smooth quartic surfaces with 352 conics. [PDF] [Journal]
    Manuscripta math. 85, 409-417 (1994)
    • The main result of this paper is that there exist smooth quartic surfaces in P3 on which there are 352 smooth conics. Up to now the maximal number of conics that can lie on a smooth quartic surface is not known, so our number should be compared with 64, the maximal number of lines on smooth quartics.
  3. Th. Bauer, T. Szemberg:
    Abelian threefolds in (P2)3. [PDF]
    In: W. Barth et al. (eds.), Abelian Varieties (pp. 19-23). Berlin: Walter de Gruyter, 1995.
    • This paper shows that the only abelian threefolds that can be embedded into P2 x P2 x P2 are products E1 x E2 x E3 of elliptic curves. This extends a result by Hulek for abelian surfaces in P2 x P2 and complements results by Birkenhake, who studied embeddings of abelian threefolds in products of two projective spaces.
  4. Th. Bauer:
    Quartic surfaces with 16 skew conics. [PDF] [Journal]
    J. reine angew. Math. 464, 207-217 (1995)
    • This paper classifies smooth quartic surfaces in projective three-space containing 16 skew conics in terms of their abelian covers. As a consequence, it is shown that the Kummer surfaces of abelian surfaces with endomorphism ring Z[√7] embed into P3, and that the quartic surfaces obtained in this way contain exactly 432 smooth conics.
  5. Th. Bauer, W. Barth:
    Poncelet theorems. [arXiv]
    Exposition. Math. 14, 125-144 (1996)
    • This paper studies the beautiful geometry underlying classical and non-classical theorems of Poncelet-type from a modern unifying point of view.
  6. Th. Bauer, T. Szemberg:
    On tensor products of ample line bundles on abelian varieties. [PDF] [Journal]
    Math. Z. 223, 79-85 (1996)
    • The first result of this paper states that on an abelian variety a tensor product of two resp. three ample line bundles is globally generated resp. very ample. This generalizes the famous classical theorem of Lefschetz, which applies to the case of tensor powers of a single line bundle. The second part of the paper provides criteria for a tensor product of two ample line bundles to be very ample.
  7. Th. Bauer, T. Szemberg:
    Higher order embeddings of abelian varieties. [PDF] [Journal]
    Math. Z. 224, 449-455 (1997)
    • This paper provides criteria for a tensor product of ample line bundles on a abelian variety to be k-jet ample, i.e., to simultaneously generate jets of given order at a given number of points.
  8. Th. Bauer, T. Szemberg:
    Primitive higher order embeddings of abelian surfaces. [PDF] [Journal]
    Trans. Amer. Math. Soc. 349, 1675-1683 (1997)
    • To determine whether a given ample line bundle on an abelian surface satisfies a certain higher order embedding condition (k-very ampleness, k-jet ampleness) is hardest in the case where the class of the bundle is primitive. For surfaces with Picard number 1, this paper gives a complete result for such bundles.
  9. Th. Bauer:
    Smooth Kummer surfaces in projective three-space. [PDF] [Journal]
    Proc. Amer. Math. Soc. 125, 2537-2541 (1997)
    • The main result of this paper says that for any positive integer d there are smooth quartics in P3 containing 16 skew smooth rational curves of degree d. In case d=1 the statement is classical (due to Traynard) and was rediscovered by Barth and Nieto in 1994.
  10. Th. Bauer:
    Seshadri constants of quartic surfaces. [PDF] [Journal]
    Math. Ann. 309, 475-481 (1997)
    • This paper determines the possible values that the global Seshadri constant of the hyperplane bundle on a smooth quartic in P3 can have. The two submaximal cases are characterized geometrically; they occur on sets of codimension one in the space of quartics.
  11. Th. Bauer:
    On the cone of curves of an abelian variety. [arXiv] [Journal]
    Am. J. Math. 120, 997-1006 (1998)
    • When is the ample cone (or, equivalently, the cone of curves) of an abelian variety rational polyhedral? It is shown in the paper that this happens if and only if the abelian variety is isogenous to a product of mutually non-isogenous abelian varieties of Picard number one.
  12. Th. Bauer:
    Seshadri constants and periods of polarized abelian varieties. [arXiv] [Journal]
    Math. Ann. 312, 607-623 (1998)
    • First, based on studying minimal period lengths, the paper provides a lower bound on the Seshadri constant of the very general abelian variety of fixed type. This yields a new criterion for very ampleness on abelian varieties. Secondly, the paper studies Seshadri constants of Prym varieties, complementing a result of Lazarsfeld for Jacobians.
  13. Th. Bauer:
    Seshadri constants on algebraic surfaces. [arXiv] [Journal]
    Math. Ann. 313, 547-583 (1999)
    • In this paper, results in various directions for Seshadri constants on algebraic surfaces are proven. It contains in particular a complete result for abelian surfaces of Picard number one, which allows to explicitly determine their Seshadri constants.
  14. Th. Bauer, S. Di Rocco, T. Szemberg:
    Generation of jets on K3 surfaces. [PDF] [Journal]
    J. Pure Appl. Algebra 146, 17-27 (2000)
    • This paper is concerned with the problem of determining how many jets a tensor power of an ample line bundle on a K3 surface generates. The main result shows that the (k+2)-nd power is k-jet ample, except for an explicitly classified exceptional case.
  15. Th. Bauer, S. Di Rocco, T. Szemberg:
    Cyclic coverings and higher order embeddings of algebraic varieties. [arXiv] [Journal]
    Trans. Amer. Math. Soc. 353, 877-891 (2001)
    • This paper studies higher order embeddings of cyclic coverings via line bundles given by pulling back »sufficiently positive« line bundles. It relates the order of the embedding of the pullback with the order of the given line bundle and of certain rank one summands of a vector bundle that is involved.
  16. Th. Bauer, T. Szemberg:
    Local positivity of principally polarized abelian threefolds. [PDF] [Journal]
    J. reine angew. Math. 531, 191-200 (2001)
    • This paper shows that for a principally polarized abelian threefold only three values of the Seshadri constant are possible 1, 3/2, or 12/7. These values are shown to correspond to geometric situations: The polarized variety is a polarized product, or it is the Jacobian of a hyperelliptic curve, or it is generic, respectively.
  17. Th. Bauer, A. Küronya, T. Szemberg:
    Zariski chambers, volumes, and stable base loci. [arXiv] [Journal]
    J. reine angew. Math. 576, 209-233 (2004)
    • In this paper a structural result on the big cone of algebraic surfaces is proven: The big cone has a locally finite decomposition into rational locally polyhedral subcones such that in each subcone (i) the support of the negative part of the Zariski decomposition is constant, (ii) the volume function is given by a single quadratic polynomial, and (iii) the stable base loci are constant (in the interior of the subcone).
  18. Th. Bauer:
    A criterion for an abelian variety to be simple. [arXiv] [Journal]
    Arch. Math. 90, 317-321 (2008)
    • An abelian variety is simple if it does not contain any non-trivial abelian subvarieties. The purpose of this paper is to provide a criterion on the codimension-one level. Specifically, it expresses simpleness in terms of the s-invariant introduced by Cutkosky-Ein-Lazarsfeld.
  19. Th. Bauer, T. Szemberg:
    Seshadri constants on surfaces of general type. [arXiv] [Journal]
    Manuscripta Math. 126, 167-175 (2008)
    • This paper studies Seshadri constants of the canonical bundle on minimal surfaces of general type. It is concerned with the question which values between 0 and 1 can occur at arbitrary points, and it shows that small values at generic points are accounted for by the geometry of the surface.
  20. Th. Bauer, C. Schulz:
    Seshadri constants on the self-product of an elliptic curve. [arXiv] [Journal]
    Journal of Algebra 320, 2981-3005 (2008)
    • In this paper Seshadri constants on the self-product of an elliptic curve are studied. It contains explicit formulas for computing the Seshadri constants of all ample line bundles on these surfaces. The proofs use methods from the geometry of numbers.
  21. Th. Bauer, T. Szemberg:
    Seshadri constants and the generation of jets. [arXiv] [Journal]
    Journal of Pure and Applied Algebra 213, 2134-2140 (2009)
    • This paper explores the connection between Seshadri constants and the generation of jets. As is well-known, one way to view Seshadri constants is to consider them as measuring the rate of growth of the number of jets that tensor powers of a line bundle generate. The paper investigates, conversely, what one can say about the number of jets once the Seshadri constant is known.
  22. Th. Bauer:
    A simple proof for the existence of Zariski decompositions on surfaces. [arXiv] [Journal]
    J. Algebraic Geom. 18, 789-793 (2009)
    • This paper contains a a quick and simple proof of the existence and uniqueness of Zariski decompositions on surfaces. While Zariski's original proof constructs the negative part of the decomposition, the present approach is based on the idea that the positive part can be constructed from a maximality condition.
  23. Th. Bauer, S. Di Rocco, B. Harbourne, M. Kapustka, A.L. Knutsen, W. Syzdek, T. Szemberg:
    A primer on Seshadri constants. [arXiv] [Publisher link]
    Contemporary Mathematics 496, 33-70 (2009)
    • The subject of Seshadri constants witnessed quite a bit of development in recent years. This text gives an account of recent progress and discusses many open questions and examples. The idea of writing these notes originated in a workshop on Seshadri constants in Essen in February 2008.
  24. Th. Bauer, M. Funke and S. Neumann:
    Counting Zariski chambers on Del Pezzo surfaces. [arXiv] [Journal]
    Journal of Algebra 324, 92-101 (2010)
    • Zariski chambers provide a natural decomposition of the big cone of an algebraic surface into rational locally polyhedral subcones that are interesting from the point of view of linear series. This paper presents an algorithm that allows to effectively determine Zariski chambers when the negative curves on the surface are known.
  25. Th. Bauer, T. Szemberg;:
    On the Seshadri constants of adjoint line bundles. [arXiv] [Journal]
    Manuscripta Math. 135, 215-228 (2011)
    • This paper is concerned with a natural question on Seshadri constants: What are the possible values? While in general every positive rational number appears as a local Seshadri constant of some ample line bundle, the paper provides various bounds and restrictions for line bundles that are adjoints of nef bundles.
  26. Th. Bauer, M. Caibar and G. Kennedy:
    Zariski decomposition: a new (old) chapter of linear algebra. [arXiv] [Journal]
    American Math. Monthly, Vol. 119, No. 1, 25-41 (2012)
    • This paper originated in the observation that the concept of Zariski decomposition is in essence purely within the realm of linear algebra. In the paper, Zariski decomposition is therefore formulated as a theorem in linear algebra and a linear algebraic proof is presented.
  27. Th. Bauer, M. Funke:
    Weyl and Zariski chambers on K3 surfaces. [arXiv] [Journal]
    Forum Mathematicum 24, 609-625 (2012)
    • The big cone of every K3 surface admits two natural chamber decompositions: the decomposition into Zariski chambers, and the decomposition into simple Weyl chambers. This paper studies the mutual relationship of these decompositions: When do they coincide? Which inclusions between chambers occur? In particular, the surprising fact is established that -- even though the decompositions themselves may differ -- the number of Zariski chambers always equals the number of simple Weyl chambers.
  28. Th. Bauer, C. Bocci, S. Cooper, S. Di Rocco, M. Dumnicki, B. Harbourne, K. Jabbusch, A. L. Knutsen, A. Küronya, R. Miranda, J. Roe, H. Schenck, T. Szemberg, Z. Teitler:
    Recent developments and open problems in linear series. [arXiv] [Publisher link]
    In: P. Pragacz (ed.), Contributions to Algebraic Geometry (pp. 93-140). EMS, 2012.
    • These notes originated in the very nice mini-workshop »Linear Series on Algebraic Varieties« in October 2010, at Oberwolfach. They contain a variety of interesting problems, which motivated the participants prior to the event, and examples, results and further problems which grew out of discussions during and shortly after the workshop.
  29. Th. Bauer, D. Schmitz:
    Zariski chambers on surfaces of high Picard number. [arXiv] [Journal]
    LMS Journal of Computation and Mathematics, Volume 15, 219-230 (2012)
    • In this paper an improved algorithm for the computation of Zariski chambers on algebraic surfaces is presented. The new algorithm significantly outperforms the so far available method and allows therefore to treat surfaces of high Picard number, where huge chamber numbers occur. Applications include the computation of the Zariski chambers supported by the lines on the Segre-Schur quartic.
  30. Th. Bauer, D. Schmitz:
    Volumes of Zariski chambers. [arXiv] [Journal]
    Journal of Pure and Applied Algebra 217, No. 1, 1153-164 (2013)
    • Zariski chambers have so far been studied both from a geometric and from a combinatorial perspective. In the present paper we complement the picture with a metric point of view by studying a suitable notion of chamber sizes. Our first result gives a precise condition for the nef cone volume to be finite and provides a method for computing it inductively. Our second result determines the volumes of arbitrary Zariski chambers from nef cone volumes of blow-downs. We illustrate the applicability of this method by explicitly determining the chamber volumes on Del Pezzo and other anti-canonical surfaces.
  31. Th. Bauer, B. Harbourne, A.L. Knutsen, A. Küronya, S. Müller-Stach, X. Roulleau, T. Szemberg:
    Negative curves on algebraic surfaces. [arXiv] [Journal]
    Duke Mathematical Journal, Vol. 162, No. 10, 1877-1894 (2013)
    • We obtain results on the number of irreducible curves of negative self-intersection on smooth complex projective surfaces. The only known examples of surfaces for which the self-intersection of irreducible curves is not bounded below are in positive characteristic, and the general expectation is that no examples can arise over the complex numbers. Indeed, one of the results of the paper shows that the idea underlying the examples in positive characteristic cannot produce examples over the complex number field.
  32. Th. Bauer, T. Szemberg:
    The effect of points fattening in dimension three. [arXiv]
    In: Ch. D. Hacon, M. Mustata, M. Popa (eds.), Recent Advances in Algebraic Geometry (pp. 1-12). London Mathematical Society Lecture Note Series: 417, Cambridge Univ. Press, 2014.
    • There has been increased recent interest in understanding the relationship between the symbolic powers of an ideal and the geometric properties of the corresponding variety. While a number of results are available for the two-dimensional case, the higher-dimensional case is largely unexplored. In the present paper we study a natural conjecture arising from a result by Bocci and Chiantini. As a first step towards understanding the higher-dimensional picture, we show that this conjecture is true in dimension three. Also, we provide examples showing that the hypotheses of the conjecture may not be weakened.
  33. Thomas Bauer, Sandra Di Rocco, Brian Harbourne, Jack Huizenga, Anders Lundman, Piotr Pokora, Tomasz Szemberg:
    Bounded negativity and arrangements of lines. [arXiv]
    Int. Math. Res. Notices 2015 (19): 9456-9471. [Journal]
    • The Bounded Negativity Conjecture predicts that for any smooth complex surface X there exists a lower bound for the self-intersection of reduced divisors on X. This conjecture is open. It is also not known if the existence of such a lower bound is invariant in the birational equivalence class of X. In the present note we introduce certain constants H(X) which measure in effect the variance of the lower bounds in the birational equivalence class of X. We focus on rational surfaces and relate the value of H(P^2) to certain line arrangements.
  34. Thomas Bauer, Sándor J Kovács, Alex Küronya, Ernesto Carlo Mistretta, Tomasz Szemberg, Stefano Urbinati:
    On positivity and base loci of vector bundles. [arXiv]
    Eur. J. Math. 1, No. 2, Article ID 38, 229-249 (2015) [Journal]
    • The aim of this note is to shed some light on the relationships among some notions of positivity for vector bundles that arose in recent decades. Our purpose is to study several of the positivity notions studied for vector bundles with some notions of asymptotic base loci that can be defined on the variety itself, rather than on the projectivization of the given vector bundle. We relate some of the different notions conjectured to be equivalent with the help of these base loci, and we show that these can help simplify the various relationships between the positivity properties present in the literature. In particular, we define augmented and restricted base loci B+(E) and B-(E) of a vector bundle E on the variety X, as generalizations of the corresponding notions studied extensively for line bundles. As it turns out, the asymptotic base loci defined here behave well with respect to the natural map induced by the projectivization of the vector bundle E.
  35. Thomas Bauer, Thorsten Herrig:
    Fixed points of endomorphisms on two-dimensional complex tori. [arXiv]
    J. Algebra 458, 351--363 (2016) [Journal]
    • In this paper we investigate fixed-point numbers of endomorphisms on complex tori. Specifically, motivated by the asymptotic perspective that has turned out in recent years to be so fruitful in Algebraic Geometry, we study how the number of fixed points behaves when the endomorphism is iterated. Our first result shows that the fixed-points function of an endomorphism on a two-dimensional complex torus can have only three different kinds of behaviours, and we characterize these behaviours in terms of the analytic eigenvalues. Our second result focuses on simple abelian surfaces and provides criteria for the fixed-points behaviour in terms of the possible types of endomorphism algebras.
  36. Thomas Bauer, Brian Harbourne, Joaquim Roé, Tomasz Szemberg:
    The Halphen cubics of order two. [arXiv]
    Collect. Math. (2017) 68, 339–357 [Journal]
    • In recent work, Roulleau and Urzúa give an implicit construction of a configuration of complex plane cubic curves, which was crucial for their results on surfaces of general type. We make this construction explicit by proving that the Roulleau-Urzúa configuration consists precisely of the Halphen cubics of a certain order, and we determine specific equations of the cubics of order 1 (which were known) and of order 2 (which are new).
  37. Thomas Bauer, Klaus Hulek, Slawomir Rams, Alessandra Sarti, Tomasz Szemberg:
    Wolf Barth (1942--2016). [arXiv] [Journal]
    Jahresber. Dtsch. Math. Ver. 119, 273–292 (2017)
    • In this article we describe the life and work of Wolf Barth who died on 30th December 2016. Wolf Barth's contributions to algebraic variety span a wide range of subjects. His achievements range from what is now called the Barth-Lefschetz theorems to his fundamental contributions to the theory of algebraic surfaces and moduli of vector bundles, and include his later work on algebraic surfaces with many singularities, culminating in the famous Barth sextic.

Preprints

  1. Thomas Bauer, Piotr Pokora, David Schmitz:
    On the boundedness of the denominators in the Zariski decomposition on surfaces. [arXiv]
    Erscheint in: Journal für die reine und angewandte Mathematik. [Journal]
    • Zariski decompositions play an important role in the theory of algebraic surfaces. For making geometric use of the decomposition of a given divisor, one needs to pass to a multiple of the divisor in order to clear denominators. It is therefore an intriguing question whether the surface has a »universal denominator« that can be used to simultaneously clear denominators in all Zariski decompositions on the surface. We prove in this paper that, somewhat surprisingly, this condition of bounded Zariski denominators is equivalent to the bounded negativity of curves that is addressed in the Bounded Negativity Conjecture. Furthermore, we provide explicit bounds for Zariski denominators and negativity of curves in terms of each other.
  2. Thomas Bauer, Carsten Bornträger:
    Nef cone volumes and discriminants of abelian surfaces. [arXiv]
    Erscheint in: Journal of Pure and Applied Algebra [Journal]
    • The nef cone volume appeared first in work of Peyre in a number-theoretic context on Fano varieties surfaces, and it was studied by Derenthal and co-authors in a series of papers on del Pezzo surfaces. The idea was subsequently extended to also measure the Zariski chambers of Del Pezzo surfaces. We start in this paper to explore the possibility to use this attractive concept to effectively measure the size of the nef cone on algebraic surfaces in general. This provides an interesting way of measuring in how big a space an ample line bundle can be moved without destroying its positivity. We give here complete results for simple abelian surfaces that admit a principal polarization and for products of elliptic curves.
  3. Thomas Bauer, Sandra Di Rocco, Brian Harbourne, Jack Huizenga, Alexandra Seceleanu, Tomasz Szemberg:
    Negative curves on symmetric blowups of the projective plane, resurgences and Waldschmidt constants. [arXiv]
    Preprint, 2016
    • The Klein and Wiman configurations are highly symmetric configurations of lines in the projective plane arising from complex reflection groups. In this paper we study the surface X obtained by blowing up P2 in the singular points of one of these line configurations. We study invariant curves on X in detail, with a particular emphasis on curves of negative self-intersection. We use the representation theory of the stabilizers of the singular points to discover several invariant curves of negative self-intersection on X, and use these curves to study Nagata-type questions for linear series on X. The homogeneous ideal I of the collection of points in the configuration is an example of an ideal where the symbolic cube of the ideal is not contained in the square of the ideal -- ideals with this property are seemingly quite rare. We use our knowledge of negative curves on X to compute the resurgence of I exactly. We also compute the asymptotic resurgence and Waldschmidt constant exactly in the case of the Wiman configuration of lines, and provide estimates on both for the Klein configuration.
  4. Thomas Bauer, Sandra Di Rocco, David Schmitz, Tomasz Szemberg, Justyna Szpond:
    On the postulation of lines and a fat line. [arXiv]
    Preprint 2017
    • In this note we show that the union of r general lines and one fat line in P3 imposes independent conditions on forms of sufficiently high degree d, where the bound on d is independent of the number of lines. This extends former results of Hartshorne and Hirschowitz on unions of general lines, and of Aladpoosh on unions of general lines and one double line.

Prof. Dr. Th. Bauer   Philipps-Universität Marburg   Fachbereich Mathematik und Informatik