The maximal rank conjecture and moduli of curves

Hypersurfaces in  projective space cut collections of points on a curve contained in that space. The maximal rank conjecture makes a prediction for the behavior of these collection of points when both the abstract curve and its immersion in projective space are general. A more refined version of the  question asks what happens when the curve is assumed to be general but the map to projective space is not .There is a predicted generic behavior. Moreover, under certain numerical conditions, the failure of that behavior gives rise to a virtual divisor in the moduli space of curves. Understanding of these divisors might allow to show that the moduli spaces of curves of genus 22 and 23 are of general type.

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