Keeping the potential difference constant , the resistance of a circuit is halved.

Why potential difference is constant?

The resistance of the resistor hasn't changed, so each coulomb of charge still needs to do the same amount of work on the resistor to flow through it. In other words, the potential difference across a component is defined as the work each coulomb of charge needs to do on the component to flow through it.

What will happen to the current when the potential difference is kept constant and the resistance of the circuit is halved?

Now, according to the question, the resistance of the circuit is halved and the potential difference is constant. It is clear from the above statement that the current becomes double.