arid, barren, dry, hot
green fields surrounded by arid desert

vast
empty, open (esp. AmE)
inhospitable
Arctic, polar
high
flat
cultural (figurative)
The town has become a cultural desert.

become, turn into, turn to
The land loses its protective cover of vegetation and soon turns into desert.

cross
He became the first person to cross the desert on foot.

stretch
The desert stretched for endless miles on all sides of us.

area, country, land, landscape, region
vast tracts of desert land

conditions
heat
sand, soil
floor, surface
plain
animal, plant

across the desert, through the desert
their journey across the desert

in the desert
cold nights in the desert

into the desert
He drove off into the desert.

Desert is used with these nouns as the subject:
luck, soldier

Desert is used with these nouns as the object:
army, nest, post, wife