Graded algebras, projective spectra and blow-ups in derived algebraic geometry

We define graded, quasi-coherent $\mathcal{O}_S$-algebras over a given base derived scheme $S$, and show that these are equivalent to derived $\mathbb{G}_{m,S}$-schemes which are affine over $S$. We then use this $\mathbb{G}_{m,S}$-action to define the projective spectrum $\mathrm{Proj} (\mathcal{B})$ of a graded algebra $\mathcal{B}$ as a quotient stack, show that $\mathrm{Proj} (\mathcal{B})$ is representable by a derived scheme over $S$, and describe the functor of points of $\mathrm{Proj} (\mathcal{B})$ in terms of line bundles. The theory of graded algebras and projective spectra is then used to define the blow-up of a closed immersion of derived schemes. Our construction will coincide with the existing one for the quasi-smooth case. The construction is done by generalizing the extended Rees algebra to the derived setting, using Weil restrictions. We close by also generalizing the deformation to the normal cone to the derived setting.