Geometry of Algebraic curves, by Arbarello, Cornalba, Griffiths, Harris, 2 volumes; Complex abelian varieties, by Lange and Birkenhake; M. Artin, Lectures on deformations of singularities.
The basic idea common to all these suggestions is to use the foundational material from Hartshorne to investigate some more specialized topics, like curves, surfaces, abelian varieties, moduli spaces, singularities, or some more general techniques like intersection theory with applications to general Riemann Roch theorems, and vanishing theorems with applications to classification questions, plus arithmetic and analytic questions.
