Subvarieties of generic hypersurfaces in any variety

Let W be a projective variety of dimension n+1, L a free line bundle on W, X in $H^0(L^d)$ a hypersurface of degree d which is generic among those given by sums of monomials from $L$, and let $f : Y \to X$ be a generically finite map from a smooth m-fold Y. We suppose that f is r-filling, i.e. upon deforming X in $H^0(L^d)$, f deforms in a family such that the corresponding deformations of $Y^r$ dominate $W^r$. Under these hypotheses we give a lower bound for the dimension of a certain linear system on the Cartesian product $Y^r$ having certain vanishing order on a diagonal locus as well as on a double point locus. This yields as one application a lower bound on the dimension of the linear system |K_{Y} - (d - n + m)f^*L - f^*K_{W}| which generalizes results of Ein and Xu (and in weaker form, Voisin). As another perhaps more surprising application, we conclude a lower bound on the number of quadrics containing certain projective images of Y.Comment: We made some improvements in the introduction and definitions. In an effort to clarify the arguments we separated the 1-filling case from the r-filling case and we gave a more detailed proof of the key lemma. The article will appear in the Math. Proc. Cambridge Philos. So

Trisecant Lemma for Non Equidimensional Varieties

The classic trisecant lemma states that if $X$ is an integral curve of \PP^3 then the variety of trisecants has dimension one, unless the curve is planar and has degree at least 3, in which case the variety of trisecants has dimension 2. In this paper, our purpose is first to present another derivation of this result and then to introduce a generalization to non-equidimensional varities. For the sake of clarity, we shall reformulate our first problem as follows. Let $Z$ be an equidimensional variety (maybe singular and/or reducible) of dimension $n$, other than a linear space, embedded into \PP^r, $r \geq n+1$. The variety of trisecant lines of $Z$, say $V_{1,3}(Z)$, has dimension strictly less than $2n$, unless $Z$ is included in a $(n+1)-$dimensional linear space and has degree at least 3, in which case $\dim(V_{1,3}(Z)) = 2n$. Then we inquire the more general case, where $Z$ is not required to be equidimensional. In that case, let $Z$ be a possibly singular variety of dimension $n$, that may be neither irreducible nor equidimensional, embedded into \PP^r, where $r \geq n+1$, and $Y$ a proper subvariety of dimension $k \geq 1$. Consider now $S$ being a component of maximal dimension of the closure of \{l \in \G(1,r) \vtl \exists p \in Y, q_1, q_2 \in Z \backslash Y, q_1,q_2,p \in l\}. We show that $S$ has dimension strictly less than $n+k$, unless the union of lines in $S$ has dimension $n+1$, in which case $dim(S) = n+k$. In the latter case, if the dimension of the space is stricly greater then $n+1$, the union of lines in $S$ cannot cover the whole space. This is the main result of our work. We also introduce some examples showing than our bound is strict

Erosion-induced massive organic carbon burial and carbon emission in the Yellow River basin, China

Soil erosion and terrestrial deposition of soil organic carbon (SOC) can potentially play a significant role in global carbon cycling. Assessing the redistribution of SOC during erosion and subsequent transport and burial is of critical importance. Using hydrological records of soil erosion and sediment load, and compiled organic carbon (OC) data, estimates of the eroded soils and OC induced by water in the Yellow River basin during the period 1950–2010 were assembled. The Yellow River basin has experienced intense soil erosion due to combined impact of natural process and human activity. Over the period, 134.2 ± 24.7 Gt of soils and 1.07 ± 0.15 Gt of OC have been eroded from hillslopes based on a soil erosion rate of 1.7–2.5 Gt yr⁻¹. Approximately 63% of the eroded soils were deposited in the river system, while only 37% were discharged into the ocean. For the OC budget, approximately 0.53 ± 0.21 Gt (49.5%) was buried in the river system, 0.25 ± 0.14 Gt (23.5%) was delivered into the ocean, and the remaining 0.289 ± 0.294 Gt (27%) was decomposed during the erosion and transport processes. This validates the commonly held assumption that 20–40% of the eroded OC would be oxidized after erosion. Erosion-induced OC redistribution on the landscape likely represented a carbon source, although a large proportion of OC was buried. In addition, about half of the terrestrially redeposited OC (49.4%) was buried behind dams, revealing the importance of dam trapping in sequestering the eroded OC. Although several uncertainties need to be better constrained, the obtained budgetary results provide a means of assessing the redistribution of the eroded OC within the Yellow River basin. Human activities have significantly altered its redistribution pattern over the past decades

Influence of Fermion Velocity Renormalization on Dynamical Mass Generation in QED3_3

We study dynamical fermion mass generation in (2+1)-dimensional quantum electrodynamics with a gauge field coupling to massless Dirac fermions and non-relativistic scalar bosons. We calculate the fermion velocity renormalization and then examine its influence on dynamical mass generation by using the Dyson-Schwinger equation. It is found that dynamical mass generation takes place even after including the scalar bosons as long as the bosonic compressibility parameter $\xi$ is sufficiently small. In addition, the fermion velocity renormalization enhances the dynamically generated mass.Comment: 6 pages, 3 figures, Chinese Physics Letter, Vol 29, page 057401(2012

Combined regenerated fibre Bragg gratings and Fabry-Perot etalons for dual strain and temperature sensing

© 2015 SPIE. A highly integrated fibre-optic sensor with regenerated fibre Bragg grating (RFBG) and a micro Fabry-Pérot (MFP) is proposed and demonstrated for simultaneous measurement of temperature and strain under high temperature (> 600°C). The MFP is fabricated by using a 157 nm fluorine gas (F2) laser to micromachine the core of a standard optical fibre. The RFBG is fabricated by regenerating a seed grating written over the Fabry-Pérot. Since the MFP and RFBG have different sensitivity coefficients, their combination can be used to realise simultaneous measurement of temperature and strain. It is believed that such a high-temperature strain sensor could find important applications in many areas where simultaneous measurement of temperature and strain under high temperature is required