I have encountered two general notions of algebraic variety when reading different texts in algebraic geometry, and wanted to ask whether they were equivalent or whether one is stronger than another.
In Gathmann's notes here, he defines a variety to be a ringed space that has a finite open cover of affine varieties, and whose diagonal is closed.
However, many other texts just define a variety to be a quasiprojective variety, which subsumes all (quasi-)affine and projective varieties.
Clearly the former definition is at least as strong as the latter, but are they equivalent? (I'm also peripherally aware of realizing varieties as certain integral, reduced, etc. schemes but I'm less familiar with this notion.) Thank you!
