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In this chapter we show a relation between higher even Gaussian maps of the canonical bundle on a smooth projective curve of genus \(g \geq 4\) and the second fundamental form of the Torelli map. This generalises a result obtained by Colombo, Pirola, and Tortora on the second Gaussian map and the second fundamental form. As a consequence, we prove that for any non-hyperelliptic curve, the Gaussian map \(\mu _{6g-6}\) is injective; hence all even Gaussian maps \(\mu _{2k}\) are identically zero for all \(k >3g-3\) . We also give an estimate for the rank of \(\mu _{2k}\) for \(g-1 \leq k \leq 3g-3.\)

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Second Fundamental Form and Higher Gaussian Maps

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  • 发表日期：

    2025年

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  • Paola Frediani

摘要

In this chapter we show a relation between higher even Gaussian maps of the canonical bundle on a smooth projective curve of genus \(g \geq 4\) and the second fundamental form of the Torelli map. This generalises a result obtained by Colombo, Pirola, and Tortora on the second Gaussian map and the second fundamental form. As a consequence, we prove that for any non-hyperelliptic curve, the Gaussian map \(\mu _{6g-6}\) is injective; hence all even Gaussian maps \(\mu _{2k}\) are identically zero for all \(k >3g-3\) . We also give an estimate for the rank of \(\mu _{2k}\) for \(g-1 \leq k \leq 3g-3.\)