An Effective Divisor in $\overline{M_{g}}$ Defined by Ramification Conditions

We define an effective divisor of the moduli space of stable curves $\overline{M_g}$, which is denoted $\overline{S^{2}W}$. Writing the class of $\overline{S^{2}W}$ in the Picard group of the moduli functor Pic$_{\text{fun}}(\overline{M_{g}})\otimes \mathbb{Q}$ in terms of the so-called Harer basis $\lambda,\delta_0,\ldots,\delta_{[g/2]}$, we prove that the relations among the coefficients of $\delta_1,\ldots,\delta_{[g/2]}$ are the same relations on coefficients as the Brill-Noether divisors. We present a result on effective divisors of $\overline{M_g}$ which could be useful to get the same relations on coefficients for other divisors. We also compute the coefficient of $\lambda$.