In this talk I will present birational maps obtained by blowing up smooth space curves of genus 11 and degree 10. We will completely characterize the loci in the Hilbert scheme $H_{g,d}^S$ where the geometry of the birational map changes. We will also examine how varying the curve in $H_{g,d}^S$ affects the rigidity of the corresponding birational map, in the sense of rank 3 fibrations.