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edited Apr 14, 2022 at 22:08

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I'm not entirely sure what would constitute an answer. But here a few simple observations. Let me focus on what you seem be interested in, namely a projective family of connected curves over smooth projective curve $f:X\to Y$. Let me also assume $X$ smooth. Let $h$ be the genus of $Y$, and $g$ the genus of the general fibre. Then from the Leray spectral sequence, Riemann-Roch and Grothendieck duality $$\chi(\mathcal{O}_X)= (1-g)(1-h) - \deg R^1f_*\mathcal{O}_X=\chi(\mathcal{O}_Y)\chi(\mathcal{O}_{X_{y_0}})+\deg f_*\omega_{X/Y}$$ A theorem of Fujita then gives an inequality $$\chi(\mathcal{O}_X)\ge \chi(\mathcal{O}_Y)\chi(\mathcal{O}_{X_{y_0}})$$ which is something. To say more, one would need to compute $\deg f_*\omega_{X/Y}$. But I think this depends on more than local information at the bad fibres, which is what I think you are asking. The reason I say this, is that this degree can be positive and variable for Kodaira surfaces, which have no bad fibres at all. [Added Comment I guess there are various things called Kodaira surfaces. The ones I have in mind are of general type, and hence algebraic. See page 220 of Compact Complex Surfaces by Barth, Hulek, Peters and Van de Ven for further details.]

I'm not entirely sure what would constitute an answer. But here a few simple observations. Let me focus on what you seem be interested in, namely a projective family of connected curves over smooth projective curve $f:X\to Y$. Let me also assume $X$ smooth. Let $h$ be the genus of $Y$, and $g$ the genus of the general fibre. Then from the Leray spectral sequence, Riemann-Roch and Grothendieck duality $$\chi(\mathcal{O}_X)= (1-g)(1-h) - \deg R^1f_*\mathcal{O}_X=\chi(\mathcal{O}_Y)\chi(\mathcal{O}_{X_{y_0}})+\deg f_*\omega_{X/Y}$$ A theorem of Fujita then gives an inequality $$\chi(\mathcal{O}_X)\ge \chi(\mathcal{O}_Y)\chi(\mathcal{O}_{X_{y_0}})$$ which is something. To say more, one would need to compute $\deg f_*\omega_{X/Y}$. But I think this depends on more than local information at the bad fibres, which is what I think you are asking. The reason I say this, is that this degree can be positive and variable for Kodaira surfaces, which have no bad fibres at all.

I'm not entirely sure what would constitute an answer. But here a few simple observations. Let me focus on what you seem be interested in, namely a projective family of connected curves over smooth projective curve $f:X\to Y$. Let me also assume $X$ smooth. Let $h$ be the genus of $Y$, and $g$ the genus of the general fibre. Then from the Leray spectral sequence, Riemann-Roch and Grothendieck duality $$\chi(\mathcal{O}_X)= (1-g)(1-h) - \deg R^1f_*\mathcal{O}_X=\chi(\mathcal{O}_Y)\chi(\mathcal{O}_{X_{y_0}})+\deg f_*\omega_{X/Y}$$ A theorem of Fujita then gives an inequality $$\chi(\mathcal{O}_X)\ge \chi(\mathcal{O}_Y)\chi(\mathcal{O}_{X_{y_0}})$$ which is something. To say more, one would need to compute $\deg f_*\omega_{X/Y}$. But I think this depends on more than local information at the bad fibres, which is what I think you are asking. The reason I say this, is that this degree can be positive and variable for Kodaira surfaces, which have no bad fibres at all. [Added Comment I guess there are various things called Kodaira surfaces. The ones I have in mind are of general type, and hence algebraic. See page 220 of Compact Complex Surfaces by Barth, Hulek, Peters and Van de Ven for further details.]

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created Apr 9, 2022 at 22:12

35.2k
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94
160

I'm not entirely sure what would constitute an answer. But here a few simple observations. Let me focus on what you seem be interested in, namely a projective family of connected curves over smooth projective curve $f:X\to Y$. Let me also assume $X$ smooth. Let $h$ be the genus of $Y$, and $g$ the genus of the general fibre. Then from the Leray spectral sequence, Riemann-Roch and Grothendieck duality $$\chi(\mathcal{O}_X)= (1-g)(1-h) - \deg R^1f_*\mathcal{O}_X=\chi(\mathcal{O}_Y)\chi(\mathcal{O}_{X_{y_0}})+\deg f_*\omega_{X/Y}$$ A theorem of Fujita then gives an inequality $$\chi(\mathcal{O}_X)\ge \chi(\mathcal{O}_Y)\chi(\mathcal{O}_{X_{y_0}})$$ which is something. To say more, one would need to compute $\deg f_*\omega_{X/Y}$. But I think this depends on more than local information at the bad fibres, which is what I think you are asking. The reason I say this, is that this degree can be positive and variable for Kodaira surfaces, which have no bad fibres at all.