Prof. Brodmann >
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About the logo
The shown surface is an embedded copy of the blown-up of the real plane with respect to two polynomials. The used computer graphics has been developed by one of my students in the framework of his Diploma Thesis in collaboration with the Department of Computer Science of our University. This part of the logo gives a hint to my second mathematical interest:
Visualization and Popularization in Mathematics
The occuring formula describes a blown-up in the language of schemes. It presents my principal mathematical interest:
Commutative Algebra and its interaction with Algebraic Geometry
What is Commutative Algebra ?
Commutative Algebra studies commutative rings, thus rings whose multiplication is a commutative operation. Examples of such rings are the Ring of integers, or polynomial rings over a field.
Commutative Algebra started to be a branch of Mathematics on its own around 1930. Today it is a major field of Algebra with an impressive richness and variety of results and many subbranches (consult the AMS classification scheme under number 13).
Commutative Algebra grew out of Algebraic Geometry and Algebraic Number Theory and yet one of its main purposes is to develop algebraic tools for these theories. In particular it became the basis of the Theory of Schemes developed by Dieudonne and Grothendieck around 1960, a theory which sets the framework for contemporary Algebraic Geometry.
Besides these more theoretic aspects, the algorithmic point of view has become a major issue in Commutative Algebra since at about 1980. In particular the technique of Groebner bases gives rise to very powerful and versatile algorithms which are used in several branches of pure Mathematics an which go on to find new applications in Science, Technology and Economics... So, today highly powered computers combined with especially designed Algebra program packages like COCOA, MACAULAY or SINGULAR became indispensable tools in many areas of Commutative Algebras.
Our recent research in Commutative Algebra and its interaction with Algebraic Geometry is focussed on questions from the following special fields (for more detailed and complete information click "Research" on the left upper corner of this page):
LOCAL COHOMOLOGY: This concept was developed by Grothendieck around 1960, initialized by Serre's fundamental work on coherent algebraic sheaves of 1955. So, local cohomology is at the core of the scheme theoretic and functorial approach to Algebraic Geometry which arose in that period. Today, Local Cohomology Theory has become an indispensable tool in almost all branches of Analytic and Algebraic Geometry as well as in Commutative and Combinatorial Algebra.
Local cohomolgy assigns to a pair (I,M), consisting of an ideal I in a noetherian ring R and an R-module M, a string of R-modules, the so called local cohomology modules of M with respect to I. Even if M is finitely generated, these local cohomology modules need not be finitely generated. On the other hand, the properties of these modules are very important in all applications of local cohomology. Therefore we study the structure of local cohomology modules.
COHOMOLOGY OF PROJECTIVE SCHEMES: This is the geometric counterpart of a particular branch of Local Cohomology Theory. In the special framework of projective varieties, Cohomology Theory of Projective Schemes opened the door to modern Algebraic Geometry with a series of fundamental results given by Serre in his work of 1955 (see above). The deepest of these results give global criteria (in terms of vanishing of cohomology groups) for local properties of coherent sheaves over projective varieties, and thus make available concepts from Complex Analysis to Algebraic Geometry over arbitrary (algebraically closed) fields.
Cohomology of Projective Schemes allows to assign to a pair (X,F), consisting of a projective scheme X (over a field for example) and a coherent sheaf F over X a string of numerical functions, the so called cohomological Hilbert functions of (X,F). These functions are a system of invariants of extraordinary significance in Projective Algebraic Geometry. We therefore study these functions and various numerical invariants related to them.
Some of these numerical invariants, as for example the Castelnuovo-Mumford regularity, are of fundamental meaning for the foundational and the computational aspect of Algebraic Geometry.
PROJECTIVE VARIETIES WITH SPECIFIED NUMERICAL INVARIANTS: The degree deg(X) of a non-degenerate irreducible variety in projective r-space (over an algebraically closed base field K) satifies the inequality deg(X)>r-dim(X), where dim(X) denotes the dimension of X. Varieties of minimal degree, thus varieties with deg(X)=r-dim(X)+1 are rather well understood. On the other hand, much less is known if deg(X)=r-dim(X)+k with k>1, even in the special cases where k=2 or k=3.
One of our aims is to study the case of "small" values of k (notably k=2 and k=3) in order to get a sufficiently complete picture of the situation from the cohomological, the homological and the geometric point of view.
The cohomological aspect includes the calculation of cohomology groups and of the so called Hartshorne-Rao module. The homological aspect includes information on the minimal free resolution of the vanishing ideal of X, notably on the occuring Betti numbers. The geometric aspect includes descriptions of X as a projection or a divisor of a variety Y with "smaller k".
Projective varieties with k =2 are called varieties of almost minimal degree. Normal varieties of almost minimal degree were classified by Fujita in 1990. An important subclass of these varieties consists of the so called Del Pezzo varieties. Our recent investigations are devoted to the classification of varieties of almost minimal degree which are not normal, using the arithmetic depth as a key invariant.
Onother very important invariant of a projective variety X in a projective r-soace is its Castelnuovo-Mumford regularity reg(X). This invariant is intimately related to the minimal free resolution of the homogeneous vanishing ideal of X and governs indeed the computational complexity of this resolution. A famous conjecture of 1984 due to Eisenbud and Goto claims that reg(X) < deg(X) + dim(X) - r. This conjecture was shown by Gruson-Lazarsfeld-Peskine in 1983 if X is a curve and by Lazarsfeld and Pinkham in 1986 if X is a smooth surface and the base field K has charactersitic 0. In many other special cases, the conjecture has been verified, but it is still open in general.
Projective varieties who satisfy the inequality reg(X) > deg(X) + dim(X) - r-1 are called varities of extremal reularity. These variety found a lot of attention and turned out to be of particular interest also in relation with the previously mentioned conjecture.
A projective variety X in projective r-space is said to be of maximal sectional regularity, it the intersection of X with a general linear subspace L of dimension r-dim(X)+1 of the ambient projective space is a curve of extremal (and hence of maximal) regularity. Varieties of maximal sectional regularity are of extremal regularity and they have a number of interesting properties. One of them is that they are either cones or else projections of rational normal scrolls.
One of our recent aims is to classify varieties of maximal sectional regularity, in notably in the surface case and thus to generalize the classification of curves of maximal regularity given by Gruson-Lazarsfekd-Peskine in 1983.
We are particularly interested in results which are independent on the characteristic of the base field K.
What do you find on this homepage ?
( > ... : click at upper left corner of this page)
- a list ot topics covered by my research
- a list of publications
> Directed Theses:
- a list of directed PhD theses
- a list of directed Diploma theses
> Talks and Teaching:
- a list of invited short courses
- a list of talks held for a general audience
- a list of invited talks given at other universities or conferences