C750 pilot reported descending below the minimum altitude depicted on a descend via STAR because of an improperly set altitude pre-select value. Crew queried ATC and were given a heading and altitude for terrain avoidance and landed normally.

Narrative

We descended below our cleared altitude during a 'descend via' arrival procedure due to a programming error on the setting of the altitude preselector.I was SIC and Pilot Monitoring on a flight from ZZZZ to ZZZ; ZZZ is our home airport. During cruise flight; the PIC and I discussed that we would be flying the ZZZZZ arrival and we both looked at the arrival plate together; because it's one we typically don't fly very often. Some time later in the flight; we were cleared to 'descend via' the arrival. The PIC was the PF (pilot flying) and he set up the AFCS (Aircraft Flight Control System) parameters for the arrival. I did not have the plate up on my iPad yet since I was preparing paperwork for Customs upon our arrival. I observed the PIC checking the restrictions in the FMS against the arrival plate on his IPad. I also observed him set the bottom altitude on the Altitude Preselect; and arm the Vertical Navigation Mode. About two minutes later I brought up the arrival plate on my own iPad and I compared the plate to the FMS; and the altitude and speed restrictions matched the approach plate. I believe I checked the 'bottom altitude' on the Altitude Preselector as I am supposed to do; but it is possible that I did not; it seems illogical that I wouldn't verify that since it is critical to be set correctly; but either way; neither the PIC nor I noticed that he had set the Altitude Preselect incorrectly. (Note: it's possible that the Preset was set correctly at this point; we don't know for sure; it could have been incorrectly changed later in the Arrival). We descended in VFR conditions on the IFR arrival; verifying altitudes and speeds against the FMS display as we proceeded; and before reaching the last fix on the arrival we prepared to configure for landing. I was SIC (pilot monitoring) working the radios; and Approach Control switched me to; when I checked in I stated '(callsign); 7;700 descending via the Arrival'; and the controller acknowledged 'roger'. We continued descending in VFR conditions. Having flown the other ZZZ arrivals many times in the past (ZZZ is my home airport); it occurred to me during the end of the arrival that a preset altitude of 5;000 feet seemed rather low; knowing the minimum vectoring altitude is usually 7;100 feet. At this point we were close to abeam the airport on a right downwind; anticipating the visual XXL. We were descending thru approximately 6;000 feet and at this point I asked the PIC 'When were we cleared to 5;000 feet?' and glancing at the AFCS panel the PIC replied to me 'Oh no; we are too low here; what happened; we were supposed to stop at 8;000 feet; and I said; 'well there is terrain coming up ahead; we need to turn or climb' and the PIC agreed with me. The PIC stopped the descent and immediately called the approach controller and stated; '(callsign) we need to turn right here for terrain clearance'. A few seconds later the controller responded '(callsign) yeah; altitude alert; turn right 20 degrees; and climb to 7;100 feet vectors for the Visual XXL'. The PIC responded 'ok; we'll turn right and we'll climb; but we have the terrain and the airport in sight for the visual'; and the controller then cleared us for the Visual Approach to XXL. We descend and landed without incident.We descended below 8;000 which was the bottom of the 'descend via' arrival because the altitude preselect was incorrectly set for 5;000. It's unclear if the incorrect altitude was set when the 'descend via' was commanded by ATC; or after the fact; but either way we busted the arrival.Cause: Poor crew coordination; the PIC loaded the arrival and briefed it out loud when setting the bottom altitude; but the SIC was not ready for that step when it occurred. We did not brief the arrival together as a crew; as is our SOP. The SIC (me) reviewed the programming independent of the PIC; but we should have briefed the arrival together with one looking at the FMS and one verifying the approach plate; which is how we usually complete this task. We were very tired after an 8 day international run of flights; and were preoccupied with the task of preparing for Customs upon arrival; so we let our guard down flying in clear VFR conditions to our home airport. CRM; the PIC is also my Supervisor and has over ten years of experience flying this airplane; compared to my one year in the plane; so from time-to-time he configures the AFCS very quickly and I am playing 'catch up' to understand what he has programmed; our SOP is to have the pilot making any change to the AFCS to verbalize that change; and the PM to confirm the change verbally; however we did not do this correctly in this particular case. Also contributing to a small degree was the approach controller not responding to my check-in radio call; when I stated we were '7;700 feet and descending via'....however it was very busy on the radios that day and we had already busted 8;000 feet at that point. Fortunately; it was VFR and we were visually maintaining traffic and terrain clearance at the time even though we on the IFR Arrival.