Seshadri slope $\K$-semistability for the blow-up of projective space along a linear subspace

RESUMO

We study a particular instance of the frame work from \cite{Grieve:CM:Line:Slope:Stab}, building on earlier work of Arezzo-et-al \cite{Arezzo:DellaVedova:LaNave}, Ross and Thomas \cite{Ross:Thomas:2007}, \cite{Ross:Thomas:2006} and others, which applies the theory of the Chow-Mumford line bundle to determine the Donaldson-Futaki invariant of deformation to the normal cone test configurations with respect to big and nef line bundles. In particular, here we consider the case of projective $n$-space blown-up along a linear subspace. Our main result arises as an application of \cite[Theorem 1.1]{Grieve:CM:Line:Slope:Stab}. It establishes $\K$-semistability of the deformation to the normal cone test configuration for the blow-up $\PP(\Osh_{\PP^s}^{\oplus r} \oplus \Osh_{\PP^s}(1)) \simeq \operatorname{Bl}_{\PP^{r-1}}(\PP^s)$ along the exceptional divisor and with respect to the tautological line bundle $\Osh_{\PP(\Osh_{\PP^s}^{\oplus r} \oplus \Osh_{\PP^s}(1))}(1)$.