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Translate the point to the origin of an affine chart and expand a local defining polynomial as $\sum a_{ij}{x}^i{y}^j$. Its lowest nonzero total degree is the multiplicity. A smooth point of a variety has multiplicity one; a double point has multiplicity two. On a point blowup of a smooth algebraic surface, the total transform is the strict transform of an algebraic subvariety plus this multiplicity times the algebraic exceptional divisor.

Blow up the distinct singular points of the integral projective curve, once at each original point, and let $C^{\prime }$ be its strict transform of an algebraic subvariety. Write $H$ for the pullback of a projective line and $E_i$ for the algebraic exceptional divisors. The multiplicity of a plane curve at a point and the intersection formula for blowing up a surface giveThe canonical divisor formula for a surface blowup gives $K=-3H+{\sum }_i{E}_i$. Applying the arithmetic adjunction formula on a smooth surface, rather than a formula requiring a smooth strict transform, yieldsand henceThe strict transform of an algebraic subvariety is still an integral projective curve. Its arithmetic genus is $h^1(C^{\prime },{\mathcal{O}}_{C^{\prime }})\ge 0$, even if some singularities remain. Therefore the plane curve singularity multiplicity bound isThe argument uses only one blowup of a smooth algebraic surface at each original singular point; it does not assume these blowups resolve every singularity.

A blowup of an algebraic variety supplies a reduced special fibre with all three requested defects. Let $B={\mathbb{A}}^3$ have coordinates $(x,y,z)$, let $p=(0,1,0)$, and let $\pi :X={\mathrm{Bl}}_{p}B\to B$. Take $Y={\mathbb{A}}^1$ and the regular mapFor an explicit construction, $X$ is the closed subvariety of ${\mathbb{A}}^3\times {\mathbb{P}}^2$, with projective coordinates $[u:v:w]$, defined byThese equations say that $(x,y-1,z)$ is proportional to $(u,v,w)$. The three standard projective charts are affine spaces ${\mathbb{A}}^3$, so $X$ is a smooth variety which is an irreducible variety embedded in a quasi-projective algebraic set. Its algebraic exceptional divisor is $E={\pi }^{-1}(p)\cong {\mathbb{P}}^2$.

If $t\ne 0$, the surface $xy=t$ misses $p$, and the blowup of an algebraic variety is an isomorphism over it. HenceIt is a smooth variety, an affine variety, an irreducible variety, and has algebraic dimension $2$.

For $t=0$, the fibre of a morphism has three irreducible components: the strict transform of an algebraic subvariety $\widetilde{V(x)}$, the unchanged plane $V(y)$, and the algebraic exceptional divisor $E\cong {\mathbb{P}}^2$. The centre $p$ lies on $V(x)$ but not on $V(y)$; this is why all three components have multiplicity one. To verify reducedness and the crossing directly near $E$, use the chart $v\ne 0$, puttingHere $x=ab$, $y=1+b$, $z=bc$, and $f=ab(1+b)$. Near $b=0$, the factor $1+b$ is a unit, so the fibre of a morphism is $ab=0$, a reduced union of two crossing planes. The $u$-chart gives $f=x(1+xv)$, whose second factor is a unit along $E$; the $w$-chart gives $f=zu(1+zv)$ and the same reduced crossing. Away from $E$, reducedness follows from $xy=0$ in $B$. Thus the scheme-theoretic fibre of a morphism is itself reduced.

It has singular points of an algebraic variety already along $V(x,y)$, which misses $p$: locally its equation is $xy=0$, with Zariski tangent space of dimension of a vector space $3$ but local dimension of an algebraic variety $2$. Finally, $E$ is a closed subvariety of $X_0$. A closed subvariety of an affine variety is an affine variety, whereas ${\mathbb{P}}^2$ is not: all its global regular functions are constant, which cannot be the coordinate ring of a positive-dimensional affine variety. Consequently $X_0$ is singular, reducible, and nonaffine, as required. Choosing the centre off the intersection of the original planes avoids introducing a nonreduced special fibre of a morphism.

For a blowup of an algebraic variety $\pi :\tilde{X}\to X$ with centre $Z$, the strict transform of an algebraic subvariety of a subvariety $Y$ not contained in $Z$ is the closure of ${\pi }^{-1}(Y\setminus Z)$. It excludes components supported entirely in the algebraic exceptional divisor.