Logarithmic Hilbert schemes of points on a curve

Weighted blow-ups are a birational transformation that naturally appears in moduli spaces. One instance where this happens, is when studying the logarithmic Hilbert scheme of points on a curve C equipped with a log structure. Today we will give a quick introduction to both weighted blow-ups and logarithmic Hilbert schemes of points on curves. Then we will focus our attention on the examples of two and three points on (P1|0) and will describe the wall crossings between the classical Hilbert scheme of points and the logarithmic ones via weighted blow-ups.