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The hosts dominated the first half and Juan Cuadrado should have done better when he drove into the box and tricked his way past a defender, before scuffing an effort across goal.
On the hour-mark, Alberto Aquilani came close for Viola with a low strike that was kept out by Perin.
Moments later, the visitors were inches away from taking the lead when Mauricio Pinilla saw his overhead kick bounce just over the bar.
Genoa's hopes of getting something out of the match took a blow in the 72nd minute when Facundo Roncaglia was sent off for picking up his second yellow card of the afternoon.
In stoppage time, Filippo Bernardeschi came close to winning the match for Fiorentina, but he saw his header saved by the the impressive Perin as Genoa held on to claim a point.