Consider the moduli stack of quadric plane curves upto projective equivalence: $\mathcal{X} =[\mathbb{P}^5/PGL(3)]$. How can I describe the tangent space of this stack? I have read somewhere a while back that the tangent space of an algebraic stack should be a cochain complex in degrees $0,1$, but I'm not sure how to give an explicit presentation, or even where this cochain complex lives (maybe in $D^{[0,1]}(\mathcal{X})$?).
How can I describe the tangent space of this quotient stack, $[\mathbb{P}^5/PGL(3)]$?
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algebraic-geometry
quotient-spaces
algebraic-stacks
tangent-bundle