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Suppose $C$ is a (non-singular) compact Riemann surface of genus $g$ and with $n$ (distinct) marked points $p_1,\ldots,p_n$. If we assume the stability condition ($2-2g-n<0$), then it is proved in "Geometry of Algebraic Curves (vol 2)" by Arbarello et al that the cohomology group $H^1(C,T_C(-p_1-\ldots-p_n))$ parametrizes the infinitesimal deformations of the curve $C$ with $n$ marked points. Here, $T_C$ is the holomorphic tangent bundle of $C$.

Is there a similar interpretation for the cohomology group $H^1(C,T_C(- D))$, where $D$ is an arbitrary effective divisor on $C$ (maybe with the additional assumption that $2-2g-\deg D<0$) with some points occurring with multiplicity (e.g. $D = 2p$, with a point $p\in C$)?

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Given a collection of positive numbers $\{n_i\}_{i=1}^N$, you can consider the moduli space $\mathcal{M}$ parametrizing compact Riemann surfaces $C$ of genus $g$ together with $N$ marked points $p_1,\dots,p_N$ and an $(n_i-1)$-jet of a coordinate at $p_i$ for every $i$. For instance, a $0$-jet of a coordinate at $p_i$ is no extra data and a $1$-jet of a coordinate is a trivialization of $T^*_{C, p_i}$.

Let $D = \sum_i n_i p_i$. Then the tangent space at a given point in $\mathcal{M}$ is given by $H^1(C, T_C(-D))$ (you should probably assume that $2 - 2g - \deg(D) < 0$).

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