Jens Piontkowski's Homepage

Heinrich-Heine Universität Düsseldorf
Mathematisches Institut
Algebraische Geometrie
Universitätsstr. 1
D-40225 Düsseldorf
Germany

Books

with S. Müller-Stach Elementare und algebraische Zahlentheorie. Vieweg Verlag 2007.
Abstract: The books covers the basic topics of number theory with an emphasis on the computability. For the tables of contents look here.
with G. Fischer Ruled varieties. An Introduction to Algebraic Differential Geometry. Vieweg 2001.
Abstract: Ruled varieties are unions of a family of linear spaces. They are objects of algebraic geometry as well as differential geometry, especially if the ruling is developable. This book is an introduction to both aspects, the algebraic and differential one. Starting from very elementary facts, the necessary techniques are developed, especially concerning Grassmannians and fundamental forms in a version suitable for complex projective algebraic geometry. Finally, this leads to recent results on the classification of developable ruled varieties and facts about tangent and secant varieties. Compared to many other topics of algebraic geometry, this is an area easily accessible to a graduate course.

Publications in Applied Mathematics

Pricing small tariffs in German health insurance. preprint.
Abstract: Traditionally, in German health insurance, a small tariff with volatile claims is priced using a similar large tariff as a reference. However, legislative changes and market forces have led to a fragmentation of tariffs. As a result, such a large similar tariff is often no longer available. Here we propose a statistical model that combines the data of several small tariffs to derive a common relative claim inflation as well as the expected claims of these tariffs in the future, thus enabling stable pricing for these tariffs.
Forecasting Health Expenses Using a Functional Data Model. Ann. Actuar. Sci. 14, 72-82 (2020).
Abstract: Traditionally, actuaries make their predictions based on simple, robust methods. Stochastic models become increasingly popular because they can enrich the point estimates with error estimates or even provide the whole probability distribution. Here, we construct such a model for German inpatient health expenses per age using the functional data approach. This allows us to see in which age groups the expenses change the most and where predictions are most uncertain. Jumps in the derived model parameters indicate that three years might be outliers. In fact, they can be explained by changes in the reimbursement system and must be dealt with. As an application we compute the probability distribution of the total health expenses in the upcoming years.
Über die Anspringwahrscheinlichkeit des Auslösenden Faktors in der Krankenversicherung. J. ZVersWiss 104, 353-364 (2015).
Zusammenfassung: Im Gegensatz zur traditionellen Lebensversicherung ist in der privaten Krankenversicherung die Mindestverzinsung der Sparbeiträge nicht über die gesamte Vertragslaufzeit garantiert, sondern kann bei einer Beitragsanpassung geändert werden, wenn abzusehen ist, dass das Versicherungsunternehmen die garantierte Höhe mittelfristig nicht mehr erzielen kann. Dies reduziert das Zinsrisiko eines Krankenversicherers deutlich. Allerdings darf eine Beitragsanpassung nur dann stattfinden, wenn der Auslösende Faktor angesprungen ist, d.h. wenn die für das nächste Jahr erwarteten Schäden die eingepreisten, kalkulatorischen Schäden deutlich übersteigen. Dadurch verbleibt ein Zinsrisiko bei geringer medizinischer Inflation. Um die Dauer zwischen zwei Beitragsanpassungen abschätzen zu können, werden hier die Anpassungshäufigkeiten in Abhängigkeit von der medizinischen Inflation berechnet. Dabei werden insbesondere die Zufallsschwankungen in den Schäden berücksichtigt, die durch die üblicherweise verwendete numerisch instabile Extrapolation das Ergebnis stark beeinflussen.
Claim Size Distributions in German Private Health Insurance.
Abstract:We examine the yearly claim size distribution in dependence of age group and sex. Named distributions are fitted to the data. Special attention is given to the heaviness of the tails due to its importance for the assessment of risks inherent in health insurance. Finally, we caution against using the normal approximation without an error estimate because simulations show a bad fit of the Gaussian distribution to claim distributions of groups thousand or even ten thousand strong.

23.06.2011 Version 2: Mistake in the discussion of the Binomial model for the claim-free insured at the end of Section 1 corrected. Additional minor changes.

Publications in Algebraic Geometry

On the Number of the Cusps of Rational Cuspidal Plane Curves. Experiment. Math. 16, 251-255 (2007).
Abstract: A cuspidal curve is a curve whose singularities are all cusps, i.e. unibranched singularities. The article describes computations which lead to the following conjecture:

A rational cuspidal plane plane curve of degree greater or equal to six has at most three cusps. The curves with precisely three cusps occur in three series.

Assuming the Flenner--Zaidenberg rigidity conjecture the above conjecture is verified up to degree 20.
The programms used are:
- The C program for the basic reduction of cases
- The Singular script for the Spectrum-Test (uses the libary cusp.lib).
Singular libary: cusp.lib. (Demo script) Computes invariants of irreducible plane curve singularities (cusps).
Abstract:A library for computing all (topological) numerical invariants of an irreducible plane curve singularity from a given set of numerical invariants. The considered invariants are: Puiseux exponents (two definitions), Puiseux pairs, multiplicity sequence, the generators of the semi-group.
All these invariants can be computed from the Puiseux exponents and used to compute the Puiseux exponents. In addition, the delta--invariant, the resolution graph, and the spectrum of the singularity can be computed from the Puiseux exponents.
The algorithms for the computation of the spectrum is taken from [Saito, K: Exponents of an irreducible plane curve singularity. math.AG/0009133 ]. The remaining algorithms are contained in the book [deJong, Th. and G. Pfister: Local Analytic Geometry. Vieweg 2000].
Theta-Characteristics on Singular Curves. J. London Math. Soc. 75, 479-494 (2007).
Abstract: On a smooth curve a theta-characteristic is a line bundle L whose square is the canonical line bundle \omega. The equivalent condition Hom(L,\omega)=L generalizes well to singular curves, as applications show. More precisely, a theta-characteristic is a torsion-free sheaf \calF of rank1 with Hom(\calF,\omega)= \calF. If the curve has non ADE-singularities then there are infinitely many theta-characteristics. Therefore, theta-characteristics are distinguished by their local type. The main purpose of this article is to compute the number of even and odd theta--characteristics (i.e. \calF with h^0(C,\calF)=0 resp. h^0(C,\calF)=1 modulo 2) in terms of the geometric genus of C and certain discrete invariants of the fixed local type.
Self-dual modules over local rings of curve singularities. J. Pure Appl. Algebra. 207, 327-339 (2006).
Abstract: Let C be a reduced curve singularity. C is called of finite self--dual type if there exist only a finitely many isomorphism classes of indecomposable, self--dual, torsion--free modules over the local ring of C. In this paper it is shown that the singularities of finite self--dual type are those which dominate a simple plane singularity.
Linear Symmetric Determinantal Hypersurfaces. Michigan Math. J. 54, 117-146 (2006). Part of the Habilitationsschrift.
Abstract: Linear symmetric determinantal hypersurfaces are hypersurfaces whose equation is the determinant of a symmetric matrix whose entries are linear forms. Here we study which cubic and quartic surfaces in P^3 can occur as such hypersurfaces. They are all singular; the possible combinations of singularities are listed.
Topology of the Compactified Jacobians of Singular Curves. Math. Z. 255, 195-226 (2007). (with shortened tables and without Section 4); Part of the Habilitationsschrift.
Abstract: We compute the Euler number of the compactified Jacobian of a curve whose minimal unibranched normalization has only plane irreducible singularities with characteristic Puiseux exponents (p,q), (4,2q,s), (6,8,s), or (6,10,s). Further, we derive a combinatorial method to compute the Betti numbers of the compactified Jacobian of an unibranched rational curve with singularities like above. Some of the Betti numbers can be stated explicitly.
Compactified Jacobians and Symmetric Determinantal Hypersurfaces. Habilitationsschrift.
Abstract: In the first part of the thesis the Betti and Euler numbers of the compactified Jacobian of singular curves with certain singularities are computed. With the help of these numbers one can count rational curves on K3-surfaces. Further, they determine the multiplicity of the delta-constant stratum in the semi-universal deformation of the singularity.
In the second part conditions are given whether a surface in projective 3-space occurs as the determinant of a symmetric matrix with linear entries. In particular, such cubics and quartics are classified.
Affinely Smooth Developable Varieties of Low Gauss Rank. Preprint.
Abstract: We study projective varieties whose image of Gauss map has dimension less or equal to four and which are smooth outside the hyperplane at infinity. We describe their geometric structure, and show in particular that they are uniruled by linear spaces which are larger than a priori expected.(Extended Version containing the structure theorem for Gauss rank 4 ).
Developable Varieties of Gauss rank 2. Inter. J. Math. 13, 93-110 (2002).
Abstract: We classify projective varieties whose Gauss map has a two dimensional image. A global geometric description of the seven types is given.
Developable Varieties with all Singularities at Infinity. manuscripta math. 106, 75-99 (2001).
Abstract: A projective variety is called developable if the image of its Gauss map has a smaller dimension than the variety itself. Developable varieties are always singular, and requiring that all singularities lie in a hyperplane puts a severe restriction on them. Here we refine a theorem of Wu and Zheng stating that such varieties are the union of cones if the dimension of the Gauss image is less than or equal to four. Afterwards we study their singular locus. Finally, we describe the geometry of such varieties whose Gauss image has dimension two.
with A. Van de Van The Automorphism Group of Linear Sections of the Grassmannians G(1,N) Documenta Math. 4, 623-664 (1999).
Abstract: The Grassmannians of lines in projective N-space, G(1,N), are embedded by way of the Plücker embedding in the projective space $\PP(\bigwedge^2\CC^{N+1})$. Let H^l be a general l-codimensional linear subspace in this projective space.
We examine the geometry of the linear sections G(1,N)\cap H^l by studying their automorphism groups and list those which are homogeneous or quasihomogeneous.
Eine Normalform für Kurven in den Grassmannvarietäten. Dissertation (1996).
A normal form for curves in Grassmannians. manuscripta math. 89, 79-85 (1996).
Abstract: We give an elementary proof of Griffiths' and Harris normal form for curves in Grassmannians which is based differential invariants.
Eine Normalform für Kurven in den Grassmannvarietäten. Diplomarbeit (1993).