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We prove that the moduli space of polarized    K  3    surfaces of genus 11 with   n   marked points is unirational when    n  ⩽  6    and uniruled when    n  ⩽  7   . As a consequence we settle a long standing but not proved assertion about the unirationality of    M   11  ,  nfor    n  ⩽  6   . We also prove that the moduli space of polarized    K  3    surfaces of genus 11 with    n  ⩾  9    marked points has non‐negative Kodaira dimension.

Geometry of the moduli space of n ‐pointed K 3 surfaces of genus 11