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We study the Severi variety $V_{d,g}$ of plane curves of degree $d$ andgeometric genus $g$. Corresponding to every such variety, there is aone-parameter family of genus $g$ stable curves whose numerical invariants wecompute. Building on the work of Caporaso and Harris, we derive a recursiveformula for the degrees of the Hodge bundle on the families in question. For$d$ large enough, these families induce moving curves in $\bar{M}_g$. We usethis to derive lower bounds for the slopes of effective divisors on$\bar{M}_g$. Another application of our results is to various enumerativeproblems on $V_{d,g}$.

Linear sections of the Severi variety and moduli of curves