In a raucous debate where candidates could barely be heard over the shouting of the crowd, the accusations flew fast and the sparring did not always include policy points so much as attacks on character and personal digs at Mayor Bill de Blasio.

Assemblywoman Nicole Malliotakis, the Republican nominee, and Bo Dietl, a businessman and television personality who is running on his own “Dump the Mayor” party line, tried in various ways to suggest that New York City has declined under Mr. de Blasio.

Here are some points of fact to help sort through the noise:

A Spike in Crime?

Ms. Malliotakis rolled out what has been one of her prime lines of attack, painting a portrait of a city where crime is more rampant than Mr. de Blasio has acknowledged.

She cited the rise in felony sex crimes, saying correctly that they had gone up during his time as mayor. Misdemeanor sex crimes are also up this year. The Police Department has attributed the rise, in part, to increased reporting by sex-crime victims. But those figures were an outlier amid broad declines in other categories.