  We study the correlation between far-infared/submm dust emission and atomic
gas column density in order to derive the properties of the high Galactic
latitude, low density, Milky Way cirrus in the foreground of the Virgo cluster
of galaxies. Dust emission maps from 60 to 850 um are obtained from SPIRE
observations carried out within the Herschel Virgo Cluster Survey, complemented
by IRAS-IRIS and Planck-HFI maps. Data from the Arecibo legacy Fast ALFA Survey
is used to derive atomic gas column densities for two broad velocity
components, low and intermediate velocity clouds. Dust emissivities are derived
for each gas component and each far-infared/submm band. For the low velocity
clouds, we measure an average emissivity 0.79 +/- 0.08 times 1E-20 MJy sr^-1
cm^2 at 250 um. After fitting a modified blackbody to the available bands, we
estimated a dust absorption cross-section 0.49 +/- 0.13 times 1E-25 cm^2 H^-1
at 250 um (with dust temperature T = 20.4 +/- 1.5 K and spectral index beta =
1.53 +/- 0.17). The results are in excellent agreement with those obtained by
Planck over a much larger coverage of the high Galactic latitude cirrus (50% of
the sky vs 0.2% in our work). For dust associated with intermediate velocity
gas, we confirm earlier Planck results and find a higher temperature and lower
emissivity and cross-section. After subtracting the modelled components, we
find regions at scales smaller than 20' where the residuals deviate
significantly from the average, cosmic-infrared-background dominated, scatter.
These large residuals are most likely due to local variations in the cirrus
dust properties (and/or the dust/atomic-gas correlation) or to high-latitude
molecular clouds with average N_H2 <~ 1E20 cm^-2. We find no conclusive
evidence for intracluster dust emission in Virgo.
