Publications

File Date: September 2011
 
This is a list of my publications, with mini-abstracts.
There is also a short version without mini-abstracts.

Dissertation and Habilitationsschrift

Papers

  1. Projective images of Kummer surfaces. [PDF]
    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. Smooth quartic surfaces with 352 conics. (joint with W. Barth) [PDF]
    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. Abelian threefolds in (P2)3. (joint with T. Szemberg) [PDF]
    Abelian Varieties (Egloffstein, 1993), De Gruyter, Berlin New York, 1995, pp. 19-23.
    • 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. Quartic surfaces with 16 skew conics. [PDF]
    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. Poncelet theorems. (joint with W. Barth) [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. On tensor products of ample line bundles on abelian varieties. (joint with T. Szemberg) [PDF]
    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. Higher order embeddings of abelian varieties. (joint with T. Szemberg) [PDF] [Journal link]
    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. Primitive higher order embeddings of abelian surfaces. (joint with T. Szemberg) [PDF] [Journal link]
    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. Smooth Kummer surfaces in projective three-space. [PDF] [Journal link]
    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. Seshadri constants of quartic surfaces. [PDF] [Journal link]
    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. On the cone of curves of an abelian variety. [arXiv] [Journal link]
    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. Seshadri constants and periods of polarized abelian varieties. [arXiv] [Journal link]
    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. Seshadri constants on algebraic surfaces.
    Math. Ann. 313, 547-583 (1999) [arXiv] [Journal link]
    • In this paper, results in various directions for Seshadri constants on algebraic surfaces are proven. It contains a complete result for abelian surfaces of Picard number one, which allows to explicitly determine their Seshadri constants.
  14. Generation of jets on K3 surfaces. (joint with S. Di Rocco and T. Szemberg) [PDF] [Journal link]
    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. Cyclic coverings and higher order embeddings of algebraic varieties. (joint with S. Di Rocco and T. Szemberg) [arXiv] [Journal link]
    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. Local positivity of principally polarized abelian threefolds. (joint with T. Szemberg) [PDF] [Journal link]
    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 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. Zariski chambers, volumes, and stable base loci. (joint with A. Küronya and T. Szemberg) [arXiv] [Journal link]
    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. Zariski chambers and stable base loci.
    Tschinkel, Yuri (ed.), Mathematisches Institut, Georg-August-Universität Göttingen: Seminars Summer Term 2004. Universitätsdrucke Göttingen. 75-82 (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.
  19. A criterion for an abelian variety to be simple. [arXiv] [Journal link]
    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.
  20. Seshadri constants on surfaces of general type. (joint with T. Szemberg) [arXiv] [Journal link]
    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.
  21. Seshadri constants on the self-product of an elliptic curve. (joint with C. Schulz) [arXiv] [Journal link]
    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.
  22. Schnittstellenmodule in der Lehramtsausbildung im Fach Mathematik. (joint with U. Partheil) [PDF] [Journal link]
    Math. Semesterber. 56, 85-103 (2009)
    • This paper (written in german) is a contribution to curriculum development in the education of teachers. 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.
  23. Seshadri constants and the generation of jets. (joint with T. Szemberg) [arXiv] [Journal Link]
    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.
  24. A simple proof for the existence of Zariski decompositions on surfaces. [arXiv] [Journal link]
    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.
  25. A primer on Seshadri constants. (joint with S. Di Rocco, B. Harbourne, M. Kapustka, A.L. Knutsen, W. Syzdek, T. Szemberg) [arXiv]
    Contemporary Mathematics, Vol. 496, 2009, pp. 33-70.
    • 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.
  26. Counting Zariski chambers on Del Pezzo surfaces. (joint with M. Funke and S. Neumann) [arXiv] [Journal link]
    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.
  27. On the Seshadri constants of adjoint line bundles. (joint with T. Szemberg) [arXiv] [Journal link]
    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.
  28. Zariski decomposition: a new (old) chapter of linear algebra. (joint with M. Caibar and G. Kennedy) [arXiv] [Journal link]
    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.

    Papers accepted for publication

  29. Weyl and Zariski chambers on K3 surfaces. (joint with M. Funke) [arXiv] [Journal link]
    (To appear in Forum Math.)
    • 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.
  30. Recent developments and open problems in linear series. (joint with 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) [arXiv]
    (To appear in Contributions to Algebraic Geometry, Impanga Lecture Notes)
    • 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.
  31. Zariski chambers on surfaces of high Picard number. (joint with D. Schmitz) [arXiv]
    (to appear in: LMS Journal of Computation and Mathematics)
    • 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.

    Preprints

  32. Negative curves on algebraic surfaces. (joint with B. Harbourne, A.L. Knutsen, A. Küronya, S. Müller-Stach, T. Szemberg) [arXiv]
    Preprint, 2011.
    • 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.
  33. Volumes of Zariski chambers. (joint with D. Schmitz) [arXiv]
    Preprint, 2012
    • 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.

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