In his game as white against an unnamed opponent before the final, Luzhin is shown supposedly checkmating with Rd1-d8, which is an illegal move because his rook at d1 is pinned against his king on h1 by black's rook at c1.
In the final game, Turati lays his King down upon being checkmated. This gesture is only used as an informal way of resigning.
When talking with Natalia's father, Luzhin attempts to give an example of a good move using one of Luzin's games. Luzhin gives the placement of the white king, black king, and white's pieces, but he fails to give the position of black's pieces. Without this information, Luzhin's example is completely meaningless.
A camera operator's reflection is in the window just after Luzhin walks past a piano (about 30 minutes in).
A cameraman and camera are reflected in the car window as Luzhin prepares to go to the wedding (about the 1:33 mark).
The adjourned position is clearly winning for Black. Players of such high caliber would trivially recognize this fact. However, the movie dialogue implies that the characters have not reached this conclusion despite ample time for evaluation.