The vertical beam (Latin staticulum) of the cross usually stayed at the crucifixion site. The victim carried only the heavy cross beam (Latin patibulum).
Crucifixions were public executions near major roads. The Romans designed them to shock and warn the people.
The Place of the Skull (Hebrew and Aramaic Golgotha; Latin calvariae, “Calvary”): Most archaeologists agree that the crucifixion of Jesus happened at the site of the modern Church of the Holy Sepulchre. This is in the Christian Quarter of the old walled city of Jerusalem.
An alternative site, Gordon’s Calvary (north of the Damascus Gate), shows what the scene might have looked like. However, it only has a tomb from the 500s BC, so it is unlikely to be where Jesus was crucified and buried.