The Luna map and the Bialynicki-Birula decomposition

In (a), $I$ is an ideal, $A\mathrm{/}I=A_0$, and projection onto degree zero gives $p$ with its fixed-locus section. Hence $F$ is connected as the image of geometrically connected $X$; it is smooth by [F2]. By [F3], $E=I\mathrm{/}{I}^2$ is finite projective over $A_0$, and each graded piece $E_n$ is projective as a direct summand. Choose $A_0$-linear sections $E_n\to A_n$ of $A_n\to E_n$ for the finitely many nonzero pieces. Their images form a graded $A_0$-submodule $W\subseteq I$ with $I=W\oplus I^2$. These choices define an equivariant map ${\mathrm{Sym}}_{A_0}E\to A$.

This map is surjective by induction on positive degree: an element of $I$ differs from a lift in $W$ by a sum of products of positive-degree elements, and each product has factors of smaller degree. Degree zero is already $A_0$. Smooth geometrically connected $X$ and $F$ are geometrically integral: distinct irreducible components of a regular scheme are disjoint and open, so connectedness leaves one. Thus $A$ and $A_0$ are domains. Locally on $F$, $E$ is free of rank $\dim X-\dim F$; the symmetric algebra is a polynomial domain of dimension $\dim X$, and its surjection to the domain $A$ has prime kernel of height zero, hence zero. These local isomorphisms give $A\cong {\mathrm{Sym}}_{A_0}E$ and the vector-bundle assertion.

For the strictly positive tangent case, the tangent space of $F$ at $x$ is the zero-weight space, hence zero; smooth connected $F$ is therefore the single rational point $x$. The algebra has $A_0=k$ and positive grading. Every homogeneous lift of a cotangent basis generates $A$ by the same induction as step 2.1 and gives a surjection from a polynomial algebra on $\dim X$ variables; dimension and integrality make it an isomorphism. This proves the Luna assertion. The same argument also applies when positivity is assumed only at the fixed point: inject $A$ into its regular local ring at $x$; [F4] and the weight decomposition of ${\mathfrak{m}}^j\mathrm{/}{\mathfrak{m}}^{j+1}$ show that every nonconstant homogeneous element has positive degree, so $A_0=k$. For an equivariant morphism with invertible differential, choose homogeneous cotangent lifts on the target and pull them back. They are homogeneous basis lifts on the source, so both resulting polynomial-algebra maps are isomorphisms, and hence so is the morphism.

In (b), choose an invariant affine neighborhood $U$ of each fixed rational point $x$. An orbit with limit $x$ stays in $U$: if its original point were in the invariant closed complement, so would its limit. Thus $X(x)$ is the affine concentrator $U(x)$, independent of the neighborhood by the functor represented in [F1]; its realization is closed in $U$, hence locally closed in $X$. It is smooth by [F1]. The defining ideal kills negative tangent directions and the zero directions from the target point, leaving exactly $T_x^{+}X$. The action extends to ${\mathbf{A}}^1$, contracts all its points to $x$, and has only this fixed point. This also proves connectedness: the image of an orbit extension is connected and meets $x$, so every geometric point belongs to the component of $x$. Step 3.1 therefore identifies $X(x)$ equivariantly with $T_x^{+}X$.

Properness extends each geometric orbit map over zero by the valuative criterion, and its limit is fixed. Hence the finitely many cells are disjoint and cover $X$. Geometric integrality of $X$ implies that exactly one cell is dense: one of their finitely many closures must be $X$, and two disjoint dense locally closed subsets would give disjoint nonempty opens. This dense locally closed cell is open. It has dimension $\dim X$, so all tangent weights there are positive. Apply the same argument to the reciprocal action to obtain a unique point with all tangent weights negative. Its attracting cell for the original action has dimension zero and, by step 4.1, is just that point. Conversely a point with zero-dimensional attracting cell has no positive tangent weights and has no zero weights because the fixed scheme is finite and smooth, so all its weights are negative; uniqueness follows from the reciprocal action. This proves all the decomposition and extremal-cell claims.