framed bundles; affine bundles; quot scheme Cox rings; algebraic varieties; homogeneous spaces; graded algebras and rings; line bundles; toric varieties; geometric invariant theory; actions of groups; algebraic surfaces; Mori Dream Spaces; Zariski decompositions; Manin's conjecture; Hasse principle; Brauer-Manin obstructions; del Pezzo surfaces; \(K3\) surfaces; Enriques surfaces; GKZ decompositions; GALE transformations; flag varieties; combinatorial methods in algebraic geometry 