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We prove that the moduli spaces of curves of genus 22 and 23 are of generaltype. To do this, we calculate certain virtual divisor classes of small slopeassociated to linear series of rank 6 with quadric relations. We then developnew tropical methods for studying linear series and independence of quadricsand show that these virtual classes are represented by effective divisors.

The Kodaira dimensions of $\overline{\mathcal{M}}_{22}$ and $\overline{\mathcal{M}}_{23}$