Resolution of singularities is one of the central and difficult problems in algebraic geometry. To date, most general approach to this problem is via algorithm in nature. In this talk, we will present a gentle discussion on resolution of singularities via geometric method.

As an important example, we will discuss the following conjecture in a leisurely way. Let g>0 and X be the moduli of genus g stable maps into the projective space P^n. Then, for every g > 0, there is a smooth moduli stack M admitting a dominating morphism X ---> M; further, there is another smooth moduli stack M' admitting a birational dominating morphism M' ---> M such that if we pullback X to M', then the resulting stack X' enjoys the following when d > 2g-2:
(1) the main component of X' is smooth;
(2) the entire stack X' has at worst normal crossing singularities.

We will explain in this talk that the above conjecture holds true when g=1 and g=2. The case of g=1 was proved by Vakil and Zinger. It was reproved by Jun Li and the speaker via algebro-geometric method along the line as formulated above. A proof of the case g=2 was initiated jointly by Jun Li and the speaker and it is now being completed by the speaker, Jun Li and Jingchen Niu.