In this talk, we will discuss the geometry and topology of singular curves. This is an interesting topic that stems from Kodaira's work originally. My talk contains two parts. In the first one, concentrating on some invariants defined by Horikawa and Xiao, we give a new bound of these invariants, and we also use these results to classify the singular curves. In the second part, we will discuss the existence of singular curves, especially how to write down their equations according to their geometric properties. This is a joint work with Jun Lu and Shengli Tan.

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