We consider the 1D nonlinear Schrödinger equation (NLS) with focusing point nonlinearity,(0.1)i∂tψ+∂x2ψ+δ|ψ|p−1ψ=0, where δ=δ(x) is the delta function supported at the origin. In this work, we show that (0.1) shares many properties in common with those previously established for the focusing autonomous translationally-invariant NLS(0.2)i∂tψ+Δψ+|ψ|p−1ψ=0. The critical Sobolev space H˙σc for (0.1) is σc=12−1p−1, whereas for (0.2) it is σc=d2−2p−1. In particular, the L2 critical case for (0.1) is p=3. We prove several results pertaining to blow-up for (0.1) that correspond to key classical results for (0.2). Specifically, we (1) obtain a sharp Gagliardo-Nirenberg inequality analogous to Weinstein [44], (2) apply the sharp Gagliardo-Nirenberg inequality and a local virial identity to obtain a sharp global existence/blow-up threshold analogous to Weinstein [44], Glassey [17] in the case σc=0 and Duyckaerts, Holmer, & Roudenko [12], Guevara [18], and Fang, Xie, & Cazenave, [13] for 0<σc<1, (3) prove a sharp mass concentration result in the L2 critical case analogous to Tsutsumi [43], Merle & Tsutsumi [36] and (4) show that minimal mass blow-up solutions in the L2 critical case are pseudoconformal transformations of the ground state, analogous to Merle [28].