The main aim of this paper is to construct certain complex prime \mathbb{Q} -Fano 3 -folds anticanonically embedded of codimension 4 . We show that such 3 -folds can be obtained as weighted complete intersections of suitable weighted projectivizations of a 13 -dimensional affine variety \mathscr{H}_{\mathbb{A}}^{13} and its related varieties, which we call key varieties. More precisely, we prove that a prime \mathbb{Q} -Fano 3 -fold of genus 3 with three 1/2(1,1,1) -singularities in the family No. 5.4 of Takagi (2002) arises as a linear section of a weighted projectivization of \mathscr{H}_{\mathbb{A}}^{13} , and conversely, any such 3 -fold can be obtained in this way. Furthermore, by relating \mathscr{H}_{\mathbb{A}}^{13} to the C_{2} -cluster variety constructed by Coughlan and Ducat (2020), we recover their constructions of prime \mathbb{Q} -Fano 3 -folds belonging to 108 families in the Graded Ring Database. The variety \mathscr{H}_{\mathbb{A}}^{13} is constructed by adapting the description of the affine cone over the Segre variety \mathbb{P}^{2}\times \mathbb{P}^{2} , defined as the null locus of the \sharp -mapping of a 9 -dimensional quadratic Jordan algebra J of a cubic form, to a coordinatization of J with 8 parameters derived via a Peirce decomposition with respect to three fixed complementary primitive idempotents.
If two quadric surfaces are birational to each other and have no rational point, then they are isomorphic to each other.
We consider a one-dimensional Schrodinger operator with matrix-valued potential. For such an operator, the WKB construction breaks down, not only at the turning points, but also at the crossing points where the potential has multiple eigenvalues. We study the connection problem at the crossing points and compute the semiclassical asymptotics of what we call microlocal scattering matrix. We apply this result to the semiclassical resonance width arising from the molecule predissociation in quantum chemistry.
The overall goal of these lecture notes is twofold: (i) to provide an overview of the state of the art for boundary layer theory for stationary flows; (ii) to state the result, in simplified form, of a recent work, Reversal in stationary Prandtl equations by Sameer Iyer and Nader Masmoudi.
This survey is devoted to the main WKB aspects of the Isozaki-Kitada parametrix and to the applications thereof in scattering theory.
