9.11.47 The quadric cone07FD7FBC796C

Let $X=\{ Z_0Z_1=Z_2^2\} \subset \mathbb P^3$. Its base conic is the quadratic Veronese image of $\mathbb P^1$, so $n=d=2$. Its homogeneous coordinate ring $\mathbb C[\xi ^2,\xi \eta ,\eta ^2]$ is the invariant ring of the sign involution acting on the normal polynomial ring $\mathbb C[\xi ,\eta ]$, and is therefore normal; thus the base is projectively normal. In the chart

one has $\beta =\mathrm{dd}^{\mathrm{c}}\log L$. Differentiation gives

Writing $\lambda _4$ for Euclidean volume, we obtain

The transverse vanishing order on $E=\{ t=0\} $ is exactly two. For $\Psi =0$ and $k=1/2$, one has $I=1/4$ and $M=1$, so $\mu _0=\theta _0^2/2$, extended by zero at $p$, is a probability measure. The logarithmic scalar and the solution mass are $c_ a=2(1-a^2)$ for every admissible deformation, including the nonclosed ones. This exact normalization follows from the preservation of the determinant.

For $\Psi =0$ and $k(s)=1+s$, one has $I=5/6$, $M=10/3$ and $c_ b=3/5$. Put $R(x)=\sqrt{(3+2x)/5}$ and $r_0=\sqrt{3/5}$. The normalized bounded solution is

Indeed, $u'=-2/[5(1+R)]=(R-1)/(1-x)$, with a removable singularity at $x=1$. Thus $u=Q(x)-Q(0)$; it is strictly decreasing in $x$, and $u(p)=\sup _ Xu=0$. As a function of $x$ it is smooth on a neighborhood of $[0,1]$, and $x$ is the restriction of a smooth ambient function, so $u$ is the restriction of a smooth ambient function. Its prescribed measure is $\mu _1=(1+x)\theta _0^2$, of mass $10/3$, and $\mathrm{MA}^{\mathrm{ext}}_{X,\theta _0}(u)=(3/5)\mu _1$, of mass two. Formula (9.11.48) also shows how the lift of this measure vanishes along $E$.

For a real parameter $\epsilon $, the function $\Psi =\epsilon |Z_0|^2/(|Z_0|^2+|Z_1|^2+|Z_2|^2)$ is smooth on $\mathbb P^2$ and preserves $\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}\Psi >0$ for sufficiently small $|\epsilon |$. Its restriction to the conic takes the values $\epsilon $ and zero at $[1:0:0]$ and $[0:1:0]$. For small nonzero $\epsilon $, the logarithmic residual is therefore nonconstant on $E$ and does not descend. Deformations supported away from $p$, including nonclosed ones, leave that residual and the full descended solution unchanged.