Let ${\rm Hilb}^{6t-3}(\mathbb{P}^3)$ be the Hilbert scheme of closed 1-dimensional subschemes of degree 6 and arithmetic genus 4 in $\mathbb{P}^3$. Let $H$ be the component of ${\rm Hilb}^{6t-3}(\mathbb{P}^3)$ whose generic point corresponds to a canonical curve, that is, a complete intersection of a quadric and a cubic surface in $\mathbb{P}^3$. Let $F$ be the vector space of linear forms in the variables $z_1, z_2, z_3, z_4$. Denote by $F_d$ the vector space of homogeneous forms of degree $d$. Set $X = \{(f_2,f_3)\}$ where $f_2 \in \mathbb{P}(F_2)$ is a quadric surface, and $f_3 \in \mathbb{P}(F_3/f_2 \cdot F)$ is a cubic modulo $f_2$. We have a rational map, $\sigma : X \cdots \rightarrow H$ defined by $(f_2,f_3) \mapsto f_2 \cap f_3$. It fails to be regular along the locus where $f_2$ and $f_3$ acquire a common linear component. Our main result gives an explicit resolution of the indeterminacies of $\sigma$ as well as of the singularities of $H$. 2000 Mathematical Subject Classification: 14C05, 14N05, 14N10, 14N15.

Persistent organic pollutants (POPs) are organic compounds that resist biochemical degradation, moving long distances across the atmosphere before deposition occurs. Our goal was to provide up-to-date data on the levels of polychlorinated dibenzo-p-dioxins/furans (PCDD/Fs) and dioxin-like polychlorinated biphenyls (DL-PCBs) in breast milk from Chilean women and to estimate the exposure of infants due to breast milk consumption. In Chile, we conducted a cross-sectional study based on methodologies proposed by the WHO, with a sample of 30 women recruited from three defined areas: 10 from the Arica Region (urban; Arica and Parinacota Region), 10 from Coltauco (rural; O’Higgins Region), and 10 from Molina (40% rural; Maule Region). High-resolution gas chromatography coupled with high-resolution mass spectrometry (HRGC/HRMS) was performed on pooled samples from each area. We calculated equivalent toxic concentrations (WHO-TEQ) based on the current WHO Toxic Equivalency Factors (TEF). The minimum and maximum values of ∑ PCDDs/Fs + DL-PCBs-TEQ were 4.317 pg TEQ/g fat in Coltauco and 6.31 pg TEQ/g fat in Arica. Molina had a total TEQ of 5.50 pg TEQ/g fat. The contribution of PCDD/Fs was approximately five-fold higher than that of DL-PCBs. The Estimated Daily Intake (EDI) of ∑ PCDDs/Fs + DL-PCBs based on the three pooled samples ranged between 6.71 and 26.28 pg TEQ/kg body weight (bw)/day, with a mean intake of 16.11 (±6.71) pg TEQ/kg bw/day in breastfed children from 0 to 24 months old. These levels were lower than those reported in international studies. Despite the fact that the observed levels were low compared to those in most industrialized countries, the detection of a variety of POPs in breast milk from Chilean women indicates the need for follow-up studies to determine whether such exposures during childhood could represent a health risk in adulthood.

In this work we get the formula (5) that it allows the explicit computation of the renormalized partition function associated with a physical system, whose states belong to a differentiable manifold of finite dimension, and whose energy function is invariant under the action of a compact Lie group. Furthermore, we extend the formula above to the infinite dimensional case using ζ-regularization technique.

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