Context :

Aims : We aimed to monitor the optical linear polarimetric signal of the magnetized , rapidly rotating M8.5 dwarf TVLM 513 - 46546 .

Methods : R - and I -band linear polarimetry images were collected with the ALFOSC instrument of the 2.56-m Nordic Optical Telescope on two consecutive nights covering about 0.5 and 4 rotation cycles in the R and I filters , respectively .
We also obtained simultaneous intensity curves by means of differential photometry .
The typical precision of the data is \pm 0.46 % ( R ) , \pm 0.35 % ( I ) in the linear polarization degree and \pm 9 mmag ( R ) , \pm 1.6 mmag ( I ) in the differential intensity curves .

Results : Strong and variable linear polarization is detected in the R and I filters , with values of maximum polarization ( p ^ { * } = 1.30 \pm 0.35 % ) similar for both bands .
The intensity and the polarimetric curves present a sinusoid-like pattern with a periodicity of \sim 1.98 h , which we ascribe to structures in TVLM 513 - 46 ’ s surface synchronized with rotation .
We found that the peaks of the intensity and polarimetric curves occur with a phase difference of 0.18 \pm 0.01 , and that the maximum of the linear polarization happens nearly half a period ( 0.59 \pm 0.03 ) after the radio pulse .
We discussed different scenarios to account for the observed properties of the light curves .

Conclusions :