Why is it the set of first order deformations equal to $H^1(X,T_{X}(-p_1, H^1(X,T_{X}(-p_1 -p_2, p_2 ..., .. -p_k))$? And what is the motivation for studying this? (I know the proof for unpointed curves, which can be found in a few sources: Hartshorne's deformation theory, Arbarello, Cornalba, Griffiths' Geometry of Algebraic Curves II etc.)
Why is it equal to $H^1(X,T_{X}(-p_1, -p_2, ..., -p_k))$? And what is the motivation for studying this? (I know the proof for unpointed curves, which can be found in a few sources: Hartshorne's deformation theory, Arbarello, Cornalba, Griffiths' Geometry of Algebraic Curves II etc.)