What's the purpose of these pull-up resistors?

NO... But they are Very useful in "De-Spiking" the Vcc as they 'smooth' out the off to on transitions by slowing very slightly the rise time of the gate pulse so that the instantaneous or edge current requirements are reduced, they are also useful to prevent a nasty oscillation that can occur in some types of rapid switching. The oscillation is caused by certain layouts and the GM or transconductance of the Mosfet. Gate resistors should be in the range of 47R to 470R for fast repetitive pulses and for slow or occasional switching they can be pretty much a whatever. In the Main the resistors serve two valid points one is a reduction in the required power supply bypassing necessary for fast repetitive switching and second is safety... If and when a Fet fails or the return path to ground fails or is lost having a current limiter between the Mosfet and your Arduino is a good idea, Ya Think?

Doc