The Moduli Spaces of Bielliptic Curves of Genus 4 with more Bielliptic Structures

Let C be an irreducible, smooth, projective curve of genus g ⩾ 3 over the complex field C. The curve C is called bielliptic if it admits a degree‐two morphism π : C → E onto an elliptic curve E such a morphism is called a bielliptic structure on C . If C is bielliptic and g ⩾6, then the bielliptic structure on C is unique, but if g=3,4,5, then this holds true only generically and there are curves carrying n≫1 bielliptic structures. The sharp bounds n ⩽ 21,10,5 exist for g=3,4,5 respectively. Let M g be the coarse moduli space of irreducible, smooth, projective curves of genus g =3,4,5. Denote byB n gthe locus of points in M g $ representing curves carrying at least n bielliptic structures. It is then natural to ask the following questions. ClearlyB g n ⊆ B g n − 1doesB g n ≠ B g n − 1hold? What are the irreducible components ofB n g ? Are the irreducible components ofB n grational? How do the irreducible components ofB n gintersect each other? Let C ∈ B g 2how many non‐isomorphic elliptic quotients can it have? Complete answers are given to the above questions in the case g =4.