Search results for "Algebraic Geometry"
showing 10 items of 356 documents
The Coble Quadric
2023
Given a smooth genus three curve $C$, the moduli space of rank two stable vector bundles on C with trivial determinant embeds in $\mathbb{P}^8$ as a hypersurface whose singular locus is the Kummer threefold of $C$; this hypersurface is the Coble quartic. Gruson, Sam and Weyman realized that this quartic could be constructed from a general skew-symmetric fourform in eight variables. Using the lines contained in the quartic, we prove that a similar construction allows to recover SU$_C(2, L)$, the moduli space of rank two stable vector bundles on C with fixed determinant of odd degree L, as a subvariety of $G(2, 8)$. In fact, each point $p \in C$ defines a natural embedding of SU$_C(2, \mathca…
Integrable systems, Frobenius manifolds and cohomological field theories
2022
In this dissertation, we study the underlying geometry of integrable systems, in particular tausymmetric bi-Hamiltonian hierarchies of evolutionary PDEs and differential-difference equations.First, we explore the close connection between the realms of integrable systems and algebraic geometry by giving a new proof of the Witten conjecture, which constructs the string taufunction of the Korteweg-de Vries hierarchy via intersection theory of the moduli spaces of stable curves with marked points. This novel proof is based on the geometry of double ramification cycles, tautological classes whose behavior under pullbacks of the forgetful and gluing maps facilitate the computation of intersection…
Groups whose subgroups satisfy the weak subnormalizer condition
2019
A subgroup X of a group G is said to satisfy the weak subnormalizer condition if $$N_G(Y)\le N_G(X)$$ for each non-normal subgroup Y of G such that $$X\le Y\le N_G(X)$$ . The behaviour of generalized soluble groups whose (cyclic) subgroups satisfy the weak subnormalizer condition is investigated.
Spaces of typen on partially ordered sets
1989
This paper contains a generalized approach to incidence geometry on partially ordered sets. A difference to the usual geometrical concepts is that points may have different size. Our main result states that a large class of spaces allows lattice theoretic characterizations. Especially, a generalized version of the Veblen-Young axiom of projective geometry has a lattice theoretic equivalent, called then-generation property (which is a generalization of the ‘Verbindungssatz’). Modularity and distributivity of a lattice of subspaces are reflected in the underlying space. Finally we give specializations and examples.
Blocking sets and partial spreads in finite projective spaces
1980
A t-blocking set in the finite projective space PG(d, q) with d≥t+1 is a set $$\mathfrak{B}$$ of points such that any (d−t)-dimensional subspace is incident with a point of $$\mathfrak{B}$$ and no t-dimensional subspace is contained in $$\mathfrak{B}$$ . It is shown that | $$\mathfrak{B}$$ |≥q t +...+1+q t−1√q and the examples of minimal cardinality are characterized. Using this result it is possible to prove upper and lower bounds for the cardinality of partial t-spreads in PG(d, q). Finally, examples of blocking sets and maximal partial spreads are given.
A Group-theoretical Finiteness Theorem
2008
We start with the universal covering space $${\*M^n}$$ of a closed n-manifold and with a tree of fundamental domains which zips it $${T\longrightarrow\*M^n}$$ . Our result is that, between T and $${\* M^n}$$ , is an intermediary object, $${T\stackrel{p} {\longrightarrow} G \stackrel{F}{\longrightarrow} \*M^n}$$ , obtained by zipping, such that each fiber of p is finite and $${T\stackrel{p}{\longrightarrow}G\stackrel{F}{\longrightarrow} \*M^n}$$ admits a section.
Elementarteiler von Inzidenzmatrizen symmetrischer Blockpläne
1986
By a study of the integral code generated by the rows of the incidence matrix and its extention the following results are obtained: Let d 1,...,d V(d 1|d 2,d 2|d 3...) be the elementary divisors of the incidence matrix of a symmetric (v,n+λ, λ) design. Then d v=(n+λ)n/g.c.d. (n, λ). Moreover, if p is a prime such that p|n, p∤λ and if x p denotes the p-part of x, then (d idv+2−i) p =n p for 2≤i≤v. For projective planes it can be shown that d 1=···=d 3n−2=1, hence $$d_{n^2 - 2n{\text{ }} + {\text{ }}5} {\text{ }} = \cdots = d_{n^2 + n} = n$$ and $$d_{n^2 - n{\text{ }} + {\text{ }}1} = (n + 1)n$$ . The paper also contains some results about elementary divisors of incidence matrices G satisfyin…
k-Weakly almost convex groups and ? 1 ? $$\tilde M^3 $$
1993
We extend Cannon's notion ofk-almost convex groups which requires that for two pointsx, y on then-sphere in the Cayley graph which can be joined by a pathl1 of length ≤k, there is a second pathl2 in then-ball, joiningx andy, of bounded length ≤N(k). Ourk-weakly almost convexity relaxes this condition by requiring only thatl1 ∝l2 bounds a disk of area ≤C1(k)n1 - e(k) +C2(k). IfM3 is a closed 3-manifold with 3-weakly almost convex fundamental group, then π1∞\(\tilde M^3 = 0\).
A knot without triple bitangency
1997
It is proved that certain trefoil knot has not triple bitangency, answering thus in the negative a conjecture of S. Izumiya and W. L. Marar.
Towards Vorst's conjecture in positive characteristic
2018
Vorst's conjecture relates the regularity of a ring with the $\mathbb{A}^1$-homotopy invariance of its $K$-theory. We show a variant of this conjecture in positive characteristic.