151 research outputs found
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
Formulas for the number of (n−2)-gaps of binary objects in arbitrary dimension
AbstractIn this paper we define the notion of a gap in an arbitrary digital binary object S in a digital space of arbitrary dimension. Then we obtain an explicit formula for the number of gaps in S of maximal dimension, derive combinatorial relations for digital curves, and discuss possible applications to image analysis of digital surfaces (in particular planes) and curves
Entanglement Classification from a Topological Perspective
Classification of entanglement is an important problem in Quantum Resource
Theory. In this paper we discuss an embedding of this problem in the context of
Topological Quantum Field Theories (TQFT). This approach allows classifying
entanglement patterns in terms of topological equivalence classes. In the
bipartite case a classification equivalent to the one by Stochastic Local
Operations and Classical Communication (SLOCC) is constructed by restricting to
a simple class of connectivity diagrams. Such diagrams characterize quantum
states of TQFT up to braiding and tangling of the ``connectome.'' In the
multipartite case the same restricted topological classification only captures
a part of the SLOCC classes, in particular, it does not see the W entanglement
of three qubits. Nonlocal braiding of connections may solve the problem, but no
finite classification is attempted in this case. Despite incompleteness, the
connectome classification has a straightforward generalization to any number
and dimension of parties and has a very intuitive interpretation, which might
be useful for understanding specific properties of entanglement and for design
of new quantum resources.Comment: 18 pages, 1 figure, version accepted to PRD. Significantly updated
text following comments of referee(s). Extra references adde
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