1,714 research outputs found
Secants of Abelian Varieties, Theta functions, and Soliton Equations
This paper is a survey on relations between secant identities and soliton
equations and applications of soliton equations to problems of algebraic
geometry, i.e., the Riemann-Schottky problem and its analogues. A short
introduction into the analytic theory of theta functions is also given.Comment: 69 pages, accepted by Russian Mathematical Survey
Entanglement of four-qubit systems: a geometric atlas with polynomial compass II (the tame world)
We propose a new approach to the geometry of the four-qubit entanglement
classes depending on parameters. More precisely, we use invariant theory and
algebraic geometry to describe various stratifications of the Hilbert space by
SLOCC invariant algebraic varieties. The normal forms of the four-qubit
classification of Verstraete {\em et al.} are interpreted as dense subsets of
components of the dual variety of the set of separable states and an algorithm
based on the invariants/covariants of the four-qubit quantum states is proposed
to identify a state with a SLOCC equivalent normal form (up to qubits
permutation).Comment: 49 pages, 16 figure
The fiber of the Griffiths map for the non-hyperelliptic Fano threefolds of genus 6
Among smooth non-rational Fano 3-folds, the non-hyperelliptic Fano 3-fold
X(10) of genus 6 has the unique property to admit a non-trivial orbit of
birationally isomorphic 3-folds, inside its moduli space. Here we prove that
these orbits are, in fact, the same as the fibers of the Griffiths period map
for X(10). This leads upto the main result of the paper: The general fiber of
the period map for X(10) is a union of two irreducible families of 3-folds:
F(1) + F(2), each F(i) -- isomorphic to the Fano surface of conics of any of
its elements. As an application, we give a negative answer to a Tjurin's
conjecture: The general X(10) is birational to a quartic double solid.Comment: Duke preprint, 52 pages, LaTex 2.0
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