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Aims: We aimed to monitor the optical linear polarimetric signal of themagnetized, rapidly rotating M8.5 dwarf TVLM 513$-$46546.  Methods: $R$- and $I$-band linear polarimetry images were collected with theALFOSC instrument of the 2.56-m Nordic Optical Telescope on two consecutivenights covering about 0.5 and 4 rotation cycles in the $R$ and $I$ filters,respectively. We also obtained simultaneous intensity curves by means ofdifferential 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 asinusoid-like pattern with a periodicity of $\sim$1.98 h, which we ascribe tostructures in TVLM 513$-$46's surface synchronized with rotation. We found thatthe peaks of the intensity and polarimetric curves occur with a phasedifference of 0.18$\pm$0.01, and that the maximum of the linear polarizationhappens nearly half a period (0.59$\pm$0.03) after the radio pulse. Wediscussed different scenarios to account for the observed properties of thelight curves.

Rotational modulation of the linear polarimetric variability of the cool dwarf TVLM 513$-$46546