- Courant Research Center
Higher Order Structures in Mathematics
Mathematisches Institut
Georg-August-Universitiät Göttingen
Bunsenstr. 3-5
D-37073 Göttingen Germany - +49-551-39-7778
If R, S, T are irreducible SL3 (C)-representations, we give an easy and explicit description of a basis of the space of equivariant maps R⊗ S→ T (Theorem 3.1). We apply this method to the rationality problem for invariant function fields.... more
If R, S, T are irreducible SL3 (C)-representations, we give an easy and explicit description of a basis of the space of equivariant maps R⊗ S→ T (Theorem 3.1). We apply this method to the rationality problem for invariant function fields. In particular, we prove the rationality of the moduli space of plane curves of degree 34. This uses a criterion which ensures the stable rationality of some quotients of Grassmannians by an SL-action (Proposition 5.4).
Abstract Over the field of one element, vector bundles over n-dimensional projective spaces are considered. It is shown that all line bundles are tensor powers of the Hopf bundle and all vector bundles are direct sums of line bundles.... more
Abstract Over the field of one element, vector bundles over n-dimensional projective spaces are considered. It is shown that all line bundles are tensor powers of the Hopf bundle and all vector bundles are direct sums of line bundles. This is in complete analogy to the case of the projective line over an arbitrary classical field, but drastically simpler in comparison with projective spaces of higher dimensions.
We consider determinantal varieties X (γ) of expected codimension defined by the maximal minors of a matrix M (γ) of linear forms representing a linear map γ. Eisenbud and Popescu have conjectured that 1-generic linear maps γ have the... more
We consider determinantal varieties X (γ) of expected codimension defined by the maximal minors of a matrix M (γ) of linear forms representing a linear map γ. Eisenbud and Popescu have conjectured that 1-generic linear maps γ have the property that the syzygy ideals I (s) of all last syzygies s of X (γ) coincide with IX (γ). We prove a geometric version of this conjecture: for 1-generic linear maps γ the syzygy varieties Syz (s)= V (I (s)) of all last syzygies have the same support as X (γ).
Abstract We prove that the moduli space M _L of Lüroth quartics in P^ 2, ie the space of quartics which can be circumscribed around a complete pentagon of lines modulo the action of PGL _3 (C) is rational, as is the related moduli space... more
Abstract We prove that the moduli space M _L of Lüroth quartics in P^ 2, ie the space of quartics which can be circumscribed around a complete pentagon of lines modulo the action of PGL _3 (C) is rational, as is the related moduli space of Bateman seven-tuples of points in P^ 2.
Abstract. We apply a heuristic method based on counting points over finite fields to the Poincaré center problem. We show that this method gives the correct results for homogeneous non linearities of degree 2 and 3. Also we obtain new... more
Abstract. We apply a heuristic method based on counting points over finite fields to the Poincaré center problem. We show that this method gives the correct results for homogeneous non linearities of degree 2 and 3. Also we obtain new evidence for Żoła̧dek's conjecture about general degree 3 non linearities.
Page 1. A Quick and Dirty Irreducibility Test for Multivariate Polynomials over Fq H. -C. Graf v. Bothmer and F. -O. Schreyer CONTENTS 1. Introduction 2. Fractions of Zeros 3. Determinantal Varieties 4. Testing 5.
Abstract: We construct an exceptional sequence of length 11 on the classical Godeaux surface X which is the Z/5-quotient of the Fermat quintic surface in P^ 3. This is the maximal possible length of such a sequence on this surface which... more
Abstract: We construct an exceptional sequence of length 11 on the classical Godeaux surface X which is the Z/5-quotient of the Fermat quintic surface in P^ 3. This is the maximal possible length of such a sequence on this surface which has Grothendieck group Z^ 11+ Z/5. In particular, the result answers Kuznetsov's Nonvanishing Conjecture, which concerns Hochschild homology of an admissible subcategory, in the negative.
Abstract.—Let (V, 0) be a germ of a complete intersection variety in C n+ k, n> 0, having an isolated singularity at 0 and X be the germ of a holomorphic vector field having an isolated zero at 0 and tangent to V. We show that in this... more
Abstract.—Let (V, 0) be a germ of a complete intersection variety in C n+ k, n> 0, having an isolated singularity at 0 and X be the germ of a holomorphic vector field having an isolated zero at 0 and tangent to V. We show that in this case the homological index and the GSV-index coincide. In the case when the zero of X is also isolated in the ambient space Cn+ k we give a formula for the homological index in terms of local linear algebra.
Let k be an algebraically closed field and S= k [x0,···, xN] the homogeneous coordinate ring of the projective space PN. Consider a closed subvariety X of PN. Let SX be the homogeneous coordinate ring of X, which is a finitely generated... more
Let k be an algebraically closed field and S= k [x0,···, xN] the homogeneous coordinate ring of the projective space PN. Consider a closed subvariety X of PN. Let SX be the homogeneous coordinate ring of X, which is a finitely generated S-module with a graduation SX=⊕ d (SX) d.
Zusammenfassung We consider the syzygytableau of a spacecurve C, an invariant, which is calculated from the minimal free resolution of IC. First we give some restrictions, that are satisfied by syzygytableaus of smooth, irreducible,... more
Zusammenfassung We consider the syzygytableau of a spacecurve C, an invariant, which is calculated from the minimal free resolution of IC. First we give some restrictions, that are satisfied by syzygytableaus of smooth, irreducible, reduced spacecurves. All tableaus that satify these restrictions are enumerated for spacecurves of degree≤ 8. Then for some of these tableaus Spacecurves are constructed. We find equations for at least one spacecurve of every possible genus and degree≤ 8.
Abstract. Ellingsrud und Peskine haben 1989 gezeigt, daß es nur endlich viele Familien von Flächen nicht allgemeinen Typs im P4 gibt. Bis heute ist es nicht gelungen, diese Familien vollständig zu klassifizieren. Wir stellen die bekannten... more
Abstract. Ellingsrud und Peskine haben 1989 gezeigt, daß es nur endlich viele Familien von Flächen nicht allgemeinen Typs im P4 gibt. Bis heute ist es nicht gelungen, diese Familien vollständig zu klassifizieren. Wir stellen die bekannten Klassifikationsergebnisse vor und geben einen Überblick über die bisher bekannten Familien.
Abstract: We prove that the bounded derived category of the surface S constructed by Barlow admits a length 11 exceptional sequence consisting of (explicit) line bundles. Moreover, we show that in a small neighbourhood of S in the moduli... more
Abstract: We prove that the bounded derived category of the surface S constructed by Barlow admits a length 11 exceptional sequence consisting of (explicit) line bundles. Moreover, we show that in a small neighbourhood of S in the moduli space of determinantal Barlow surfaces, the generic surface has a semiorthogonal decomposition of its derived category into a length 11 exceptional sequence of line bundles and a category with trivial Grothendieck group and Hochschild homology, called a phantom category.
Abstract Let G= SL n (ℂ)⋉ ℂ n be the (special) affine group. In this paper we study the representation theory of G and in particular the question of rationality for V/G, where V is a generically free G-representation. We show that the... more
Abstract Let G= SL n (ℂ)⋉ ℂ n be the (special) affine group. In this paper we study the representation theory of G and in particular the question of rationality for V/G, where V is a generically free G-representation. We show that the answer to this question is positive (Theorem 6.1) if the dimension of V is sufficiently large and V is indecomposable.
Abstract Let G be one of the groups SL n (ℂ), Sp 2n (ℂ), SO m (ℂ), O m (ℂ), or G 2. For a generically free G-representation V, we say that N is a level of stable rationality for V/G if V/G× ℙ N is rational. In this paper we improve known... more
Abstract Let G be one of the groups SL n (ℂ), Sp 2n (ℂ), SO m (ℂ), O m (ℂ), or G 2. For a generically free G-representation V, we say that N is a level of stable rationality for V/G if V/G× ℙ N is rational. In this paper we improve known bounds for the levels of stable rationality for the quotients V/G. In particular, their growth as functions of the rank of the group is linear for G being one of the classical groups.
Krisentests umfassen sowohl quantitative als auch qualitative Kriterien. Durch Berücksichtigung quantitativer Kriterien sollen plausible Szenarien bestimmt werden, denen die Bank im Zuge krisenhafter Marktereignisse ausgesetzt sein kann.... more
Krisentests umfassen sowohl quantitative als auch qualitative Kriterien. Durch Berücksichtigung quantitativer Kriterien sollen plausible Szenarien bestimmt werden, denen die Bank im Zuge krisenhafter Marktereignisse ausgesetzt sein kann. Anhand qualitativer Kriterien ist abzuschätzen, ob die vorhandenen Risikodeckungsmittel potentielle Verluste abfangen können und ob Maßnahmen identifiziert werden können, die zu einer zumindest partiellen Erhaltung des Risikodeckungspotentials führen.
Abstract: Let M_ {7, n} be the (coarse) moduli space of smooth curves of genus 7 with n marked points defined over the complex field. We denote by M^ 1_ {7, n; 4} the locus of points inside M_ {7, n} representing curves carrying ag^ 1_4.... more
Abstract: Let M_ {7, n} be the (coarse) moduli space of smooth curves of genus 7 with n marked points defined over the complex field. We denote by M^ 1_ {7, n; 4} the locus of points inside M_ {7, n} representing curves carrying ag^ 1_4. It is classically known that M^ 1_ {7, n; 4} is irreducible of dimension 17+ n. We prove in this paper that M^ 1_ {7, n; 4} is rational for 0<= n<= 11.
Abstract: We prove that for a general canonical curve $ C\ subset\ mathbb {Z}^{g-1} $ of genus $ g $, the space of ${\ lceil\ frac {g-5}{2}\ rceil} $ th (last) scrollar syzygies is isomorphic to the Brill-Noether locus $ C^ 1_ {\ lceil\... more
Abstract: We prove that for a general canonical curve $ C\ subset\ mathbb {Z}^{g-1} $ of genus $ g $, the space of ${\ lceil\ frac {g-5}{2}\ rceil} $ th (last) scrollar syzygies is isomorphic to the Brill-Noether locus $ C^ 1_ {\ lceil\ frac {g+ 2}{2}\ rceil} $. Schreyer has conjectured that these scrollar syzygies span the space of all ${\ lceil\ frac {g-5}{2}\ rceil} $ th (last) syzygies of $ C $. Using Mukai varieties we prove this conjecture for genus $6$, $7$ and $8$.
Page 1. Rationality of Moduli Spaces of Plane Curves of Small Degree Christian Böhning, Hans-Christian Graf von Bothmer, and Jakob Kröker CONTENTS 1. Introduction 2. The Double Bundle Method: Algorithms 3. The Method of Covariants:... more
Page 1. Rationality of Moduli Spaces of Plane Curves of Small Degree Christian Böhning, Hans-Christian Graf von Bothmer, and Jakob Kröker CONTENTS 1. Introduction 2. The Double Bundle Method: Algorithms 3. The Method of Covariants: Algorithms 4.
Abstract: Based on a recent result of Voisin [2001] we describe the last nonzero syzygy space in the linear strand of a canonical curve C of even genus g= 2k lying on a K3 surface, as the ambient space of a k-2-uple embedded P^{k+ 1}.... more
Abstract: Based on a recent result of Voisin [2001] we describe the last nonzero syzygy space in the linear strand of a canonical curve C of even genus g= 2k lying on a K3 surface, as the ambient space of a k-2-uple embedded P^{k+ 1}. Furthermore the geometric syzygies constructed by Green and Lazarsfeld [1984] from g^ 1_ {k+ 1}'s form a non degenerate configuration of finitely many rational normal curves on this P^{k+ 1}.
Abstract: We prove that the semiorthogonal decompositions of the derived category of the classical Godeaux surface X do not satisfy the Jordan-H\" older property. More precisely, there are two maximal exceptional sequences in this... more
Abstract: We prove that the semiorthogonal decompositions of the derived category of the classical Godeaux surface X do not satisfy the Jordan-H\" older property. More precisely, there are two maximal exceptional sequences in this category, one of length 11, the other of length 9. Assuming the Noetherian property for semiorthogonal decompositions, one can define, following Kuznetsov, the Clemens-Griffiths component for each fixed maximal decomposition.
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QUANTITATIVE FINANCE COMMENTS it is most needed. As Keynes observed, stock markets behave like ill-mannered casinos, and&amp;amp;amp;amp;#x27;when the capital development of a country becomes a by-product of the activities of a... more
QUANTITATIVE FINANCE COMMENTS it is most needed. As Keynes observed, stock markets behave like ill-mannered casinos, and&amp;amp;amp;amp;#x27;when the capital development of a country becomes a by-product of the activities of a casino, the job is likely to be ill-done*([3], p 159). It is possible to suggest changes in the nature of financial instruments that will force the markets to focus less upon short-term appreciation and more upon future returns ([5],[8], pp 254-7).