1,502 research outputs found
Experimental results for the Poincar\'e center problem (including an Appendix with Martin Cremer)
We apply a heuristic method based on counting points over finite fields to
the Poincar\'e 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 Zoladek's conjecture about general degree 3 non linearitiesComment: 16 pages, 2 figures, source code of programs at
http://www-ifm.math.uni-hannover.de/~bothmer/strudel/. Added references, the
result of Example 6.2 is not new. Added two new sections on rationally
reversible systems. The 4th codim 7 component we saw only experimentally can
now also be identified geometrical
Significance of log-periodic signatures in cumulative noise
Using methods introduced by Scargle in 1978 we derive a cumulative version of
the Lomb periodogram that exhibits frequency independent statistics when
applied to cumulative noise. We show how this cumulative Lomb periodogram
allows us to estimate the significance of log-periodic signatures in the S&P
500 anti-bubble that started in August 2000.Comment: 14 pages, 7 figures; AMS-Latex; introduction rewritten, some points
of the exposition clarified. Author-supplied PDF file with high resolution
graphics is available at http://btm8x5.mat.uni-bayreuth.de/~bothmer
Geometric Syzygies of Canonical Curves of even Genus lying on a K3-Surface
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}. This proves a natural generalization of Green's
conjecture [1984], namely that the geometric syzygies should span the space of
all syzygies, in this case.Comment: 29 pages; 5 figure
On stable rationality of some conic bundles and moduli spaces of Prym curves
We prove that a very general hypersurface of bidegree (2, n) in P^2 x P^2 for
n bigger than or equal to 2 is not stably rational, using Voisin's method of
integral Chow-theoretic decompositions of the diagonal and their preservation
under mild degenerations. At the same time, we also analyse possible ways to
degenerate Prym curves, and the way how various loci inside the moduli space of
stable Prym curves are nested. No deformation theory of stacks or sheaves of
Azumaya algebras like in recent work of Hasset-Kresch-Tschinkel is used, rather
we employ a more elementary and explicit approach via Koszul complexes, which
is enough to treat this special case.Comment: 23 pages; Macaulay 2 code used for verification of parts of the paper
available at http://www.math.uni-hamburg.de/home/bothmer/m2.html and at the
end of the TeX file; v2: in section 4, we now included a proof of the main
theorem that works for all n (unconditional on the parity) that was
communicated to us by Zhi Jiang, Zhiyu Tian, and Letao Zhang. Several other
minor expository improvement
Degenerations of Gushel-Mukai fourfolds, with a view towards irrationality proofs
We study a certain class of degenerations of Gushel-Mukai fourfolds as conic
bundles, which we call tame degenerations and which are natural if one wants to
prove that very general Gushel-Mukai fourfolds are irrational using the
degeneration method due to Voisin, Colliot-Th\'{e}l\`{e}ne-Pirutka, Totaro et
al. However, we prove that no such tame degenerations do exist.Comment: 25 page
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