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 $p\in C$)?



Deformation Theoretic Interpretation of $H^1(C,T_C(-2p))$
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 $2 - 2g - \deg(D) < 0$).


