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RESEARCH PRODUCT

MR 2918162 Reviewed Van der Geer G. and Kouvidakis A. The Hodge bundle on Hurwitz spaces. Pure and Applied Mathematics Quarterly (2011) 7, no. 4, 1297 -- 1307. (Reviewer Francesca Vetro) 14H10 (14H30)

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In this paper the authors consider the Hurwitz space $H_{g, \, d}$ that parametrizes degree $d$ simple coverings of $\mathbb{P}^{1}$ with $b = 2 g - 2 + 2d$ branch points. The compactification $\bar{H}_{g, \, d}$ of this Hurwitz space is the space of admissible covers of genus $g$ and degree $d$, $f: C \rightarrow P$, where $C$ is a nodal curve and $P$ is a stable $b$-pointed curve of genus $0$. Assigning to $f: C \rightarrow P$ the stabilized model of $C$, one defines a natural map $\phi: \bar{H}_{g, \, d} \rightarrow \bar{M}_{g}$ where $\bar{M}_{g}$ denotes the moduli space of stable curves of genus $g$. The Hurwitz space $\bar{H}_{g, \, d}$ carries a natural $\mathbb{Q}$-divisor class, the Hodge class $\lambda$. This class is the pullback, under $\phi$, of the first Chern class of the Hodge bundle on $\bar{M}_{g}$. Kokotov, Korotkin and Zograf in [A. Kokotov, D. Korotkin, P. Zograf, Isomonodromic tau function on the space of admissible covers, arxiv:0912.3909v3] gave a formula for $\lambda$ on $\bar{H}_{g, \, d}$. They used analytic tools in order to prove it. In this paper the authors give an algebraic proof of this formula. Moreover they observe that such proof extends also to other cases. So, for example, they study the Hurwitz space $H_{g, \, d, \, l}$ of covers $f: C \rightarrow P$ where $C$ is smooth of genus $g$, $P = (\mathbb{P}^{1}, p_{1}, \ldots, p_{b})$ is a $b$-pointed smooth curve of genus $0$, $f$ is a morphism of degree $d$ with simple branch points $p_{i}$, for $i = 2, \ldots, b$, and one branch point $p_{1}$ over which $f$ is \'{e}tale except for one $l$-fold ramification points.