Let us suppose someone knows most of the concepts from Algebraic Geometry texts like Hartshorne. What papers should he start reading to quickly start getting a feel for research?
I find myself at such a juncture, and most papers I find on arxiv are a little too specialized for me.
Do you have an advisor? If so, they are surely better-placed to answer this question than strangers on MSE.
Having said that, one way to bridge the gap to research papers is to read books that are more specialised than the likes of Hartshorne.
Looking at my bookshelf, one good example that comes to mind is Moduli of Curves by Harris and Morrison. This is a very nice, readable book, written for an audience whose background knowledge is similar to yours, but with a lot of serious material in it. After reading this book, I felt like I could at least understand some of the questions that were being answered in research papers in this area, if not necessarily the details of the answers.
Of course, there are many other texts on different topics that you might try instead: for example, Birational Geometry of Algebraic Varieties by Kollár and Mori, Positivity in Algebraic Geometry by Lazarsfeld, An Introduction to Invariants and Moduli by Mukai... the choice is yours!