The maximal-contact mechanism fails in positive characteristic

The char-2 computation. In $k[x,y]$ with $\mathrm{char}k=2$ one has $\mathrm{\partial }(x^2)\mathrm{/}\mathrm{\partial }x=2x=0$, $\mathrm{\partial }(x^2)\mathrm{/}\mathrm{\partial }y=0$, $\mathrm{\partial }(y^2)\mathrm{/}\mathrm{\partial }x=0$ and $\mathrm{\partial }(y^2)\mathrm{/}\mathrm{\partial }y=2y=0$. Hence $\mathcal{D}(\mathcal{I})=\mathcal{I}$ by [F1], and iterating ${\mathcal{D}}^i(\mathcal{I})=\mathcal{I}$ for every $i$.

No tangent direction exists and the criterion fails. By [F2, F3] the marked ideal $(\mathcal{I},2)$ is of maximal order with support $\{0\}\mathrm{\ne }\mathrm{\varnothing }$. But ${\mathcal{D}}^1(\mathcal{I})=\mathcal{I}=(x^2,y^2)$, and every element of $(x^2,y^2)$ has order at least $2$ at the origin; hence no section of $\mathcal{D}(\mathcal{I})$ has multiplicity one, so $(\mathcal{I},2)$ admits no tangent direction at the origin, and in particular no hypersurface of maximal contact in the sense of [F4]. Equivalently, the defining criterion of maximal order fails: ${\mathcal{D}}^2(\mathcal{I})=\mathcal{I}\mathrm{\ne }{\mathcal{O}}_{{\mathbb{A}}^2}$, so the characteristic-zero equivalence between the order condition and ${\mathcal{D}}^{\mu }(\mathcal{I})={\mathcal{O}}_X$ breaks down. This refutes the first conjunct of the displayed statement.

For the second conjunct, take an algebraically closed field of characteristic two and $f=x^2+y{z}^3+z{w}^3+y^{7}w$. At a closed point its order is at most two, because its translated polynomial has coefficient $1$ on the square of the $x$-increment. It has order two exactly where $f$ and its first derivatives $z^3+y^{6}w$, $y{z}^2+w^3$, and $z{w}^2+y^7$ vanish. Substituting $(x,y,z,w)=(t^{32},t^7,t^{19},t^{15})$ makes $f$ a sum of four copies of $t^{64}$ and the three derivatives sums of two identical monomials, hence all vanish. The image curve therefore lies in the top locus; equality with the full top locus is unnecessary. Suppose a regular hypersurface germ at the origin contained this curve. Its completed equation would have a nonzero linear part $a_{x}x+a_{y}y+a_{z}z+a_{w}w$. Under substitution the linear terms have exponents $32,7,19,15$, respectively, each distinct and none a sum of at least two of these four positive weights. Thus no nonlinear monomial can cancel any nonzero linear coefficient, so the substituted series cannot vanish. A hypersurface regular at the origin must have a nonzero linear part, and this contradiction proves that no such hypersurface contains the top locus. This verifies the Narasimhan obstruction used here directly, following the polynomial and parametrization in Hauser, §14, Example 1, pp. 387–388.

Conclusion. Neither example disproves resolution of singularities in positive characteristic; both show that the characteristic-zero maximal-contact mechanism and its hypersurface criterion do not extend as stated. In particular this page's characteristic-zero theorem and its invariant cannot be quoted in positive characteristic, which is why no positive-characteristic resolution claim is made.