In the study of algebraic surfaces, the surfaces of general type form the biggest and most mysterious class. There is no hope to get a complete classification of them but some interesting results can be obtained by fixing some (topological and birational) invariants: the most important are the genus, the bigenus and the irregularity.
Among those, the first surfaces studied were those of genus 4: the first xamples come from the XIX century. Thanks to the work of many authors (mainly Enriques, Horikawa and Bauer) we have now a complete lassification of them up to bigenus 12. In particular we know that they have at least bigenus 9, and that for each value of the bigenus among 9 and 12 all minimal surface of genus 4 and this bigenus are homeomorphic. The situation changes drastically for bigenus 13: we know three examples of surfaces of general type with genus 4 and bigenus 13 which are not pairwise homeomorphic. In a joint work (in progress) with F. Catanese and W. Liu, we classify the surfaces belonging to one of these topological types, the "even" surfaces, and describe the structure of the corresponding piece of the moduli space of surfaces of general type, showing that it has one connected component but two irreducible components. I'll report on this work, showing how the two components naturally comes out from two different ways of writing the equations of a curve of genus 4 with a special Weierstrass point.
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