Small irreducible components of arc spaces in positive characteristic: component status, transfer of arc-sharpness, curve selection at arc-blunt points, and birational elimination

For an algebraic variety $X$ over a field of characteristic $p>0$ , the arc space $X_\infty$ — the scheme representing formal curves on $X$ — escapes the guarantees that hold in characteristic zero: Kolchin's irreducibility theorem fails, and the preimage $j^{-1}(\mathrm{Sing}\,X)$ of the singular locus under the truncation map $j:X_\infty\to X$ can acquire 'small' irreducible components contained in $(\mathrm{Sing}\,X)_\infty$ that have no characteristic-zero analogue. Benito, Piltant and Reguera (2022) characterized when this happens at codimension-one points through the notion of arc-sharpness and posed four linked open questions that delineate the unresolved structure theory of arc spaces in positive characteristic: (1) whether, at arc-blunt points, every small irreducible component of $j^{-1}(\mathrm{Sing}\,X)$ is an irreducible component of the whole arc space; (2) under which conditions arc-sharpness transfers along proper birational morphisms with separable residue field extensions — concretely, whether irreducibility of the arc space of a threefold in characteristic $p\ge3$ is detected by separable points in fibers of resolutions; (3) whether a curve selection lemma producing wedges through the generic arc of a small component holds at arc-blunt points; and (4) whether every valuation admits a birational model with irreducible arc space. No resolution of any of the four questions is known to date.

Background

The arc space of an algebraic variety $X$ over a perfect field $k$ is the — generally infinite-dimensional — scheme $X_\infty$ representing the functor $A\mapsto X(A[[t]])$ on $k$ -algebras: its points over a field are formal arcs, i.e. morphisms $\mathrm{Spec}\,K[[t]]\to X$ . It is the inverse limit of the jet schemes $X_m$ , which parametrize arcs truncated at order $m$ , and the projection $j:X_\infty\to X$ remembers only the starting point of an arc. Arc spaces underlie the Nash program on singularities: given a resolution of singularities (a proper birational map $\pi:Y\to X$ from a smooth variety that is an isomorphism over the smooth locus), each exceptional divisor $E$ has an associated Nash family $N_E$ , the closure in $X_\infty$ of the set of arcs whose lift to $Y$ meets $E$ , and the Nash problem asks whether these families account for all irreducible components of $j^{-1}(\mathrm{Sing}\,X)$ . In characteristic zero the theory is anchored by Kolchin's Irreducibility Theorem (from differential algebra): $X_\infty$ is irreducible whenever $X$ is. In characteristic $p>0$ this fails — the irreducible surface $V(y^p+zx^p)\subset\mathbb{A}^3_k$ has an arc space with two irreducible components, $(\mathrm{Sing}\,X)_\infty$ and the closure of its complement — and $j^{-1}(\mathrm{Sing}\,X)$ can acquire small irreducible components, meaning irreducible components contained in $(\mathrm{Sing}\,X)_\infty$ , with no characteristic-zero analogue. Benito, Piltant and Reguera (2022) built the current framework for studying them. For a point $\zeta\in X$ with $Z=\overline{\{\zeta\}}$ , the set $Z_\infty\setminus(\mathrm{Sing}\,Z)_\infty$ is irreducible in any characteristic, and its closure in $X_\infty$ , denoted $Z_\infty^\circ$ , has generic point $\zeta_\infty$ (the generic arc through $Z$ ); writing $X_1,\dots,X_c$ for the irreducible components of $X$ and $X_i^{\infty\circ}$ for the corresponding generic arc components, $X$ is called arc-sharp at $\zeta$ if $Z_\infty^\circ=X_i^{\infty\circ}$ for some $i$ or $Z_\infty^\circ\nsubseteq\bigcup_iX_i^{\infty\circ}$ , and arc-blunt at $\zeta$ otherwise. Their main results are: a necessary condition for arc-sharpness (Theorem 4.4: if $X$ admits a resolution $\pi:Y\to X$ , then $X$ arc-sharp at $\zeta$ forces $\pi^{-1}(\zeta)$ to have no separable point over $k(\zeta)$ ); an arithmetic corollary ( $X$ arc-sharp at $\zeta$ only if the multiplicity of $X$ at $\zeta$ is divisible by $p$ ); a sufficient condition via radicial (purely inseparable) morphisms (Theorem 5.5); and a complete characterization at codimension-one points (Corollary 5.8: when $\dim\mathcal{O}_{X,\zeta}=1$ , $X$ is arc-sharp at $\zeta$ if and only if $k(\bar\zeta)|k(\zeta)$ is inseparable for every point $\bar\zeta$ of the normalization above $\zeta$ ), from which the full decomposition of $X_\infty$ for surfaces follows (Corollary 5.9). Section 6 of that paper poses four questions — on the global component status of small components, the transfer of arc-sharpness along proper birational maps, curve selection at arc-blunt points, and birational elimination of small components — which remain the central open problems of this theory and are stated in the problem statement below.

Problem Statement

The source-faithful problem is the suite of four questions posed in Section 6 of Benito–Piltant–Reguera (2022), which jointly ask for a structure theory of arc spaces of positive-characteristic varieties beyond the reach of Kolchin-style methods. Throughout, $k$ is a field of characteristic $p>0$ , $X$ is an irreducible $k$ -variety, $j:X_\infty\to X$ is the order-zero truncation, and for $\zeta\in X$ with $Z=\overline{\{\zeta\}}$ , $Z_\infty^\circ$ denotes the closure in $X_\infty$ of $Z_\infty\setminus(\mathrm{Sing}\,Z)_\infty$ , with generic point $\zeta_\infty$ ; a small irreducible component of $j^{-1}(\mathrm{Sing}\,X)$ is an irreducible component contained in $(\mathrm{Sing}\,X)_\infty$ , and $X$ is arc-blunt at $\zeta$ if $Z_\infty^\circ\subseteq\overline{X_\infty\setminus(\mathrm{Sing}\,X)_\infty}$ while $\zeta$ is not a maximal (generic) point of $X$ . (1) Component status of small arcs: if $\zeta\in\mathrm{Sing}\,X$ and $X$ is arc-blunt at $\zeta$ , is every small irreducible component of the closed set $j^{-1}(\mathrm{Sing}\,X)$ necessarily an irreducible component of the whole arc space $X_\infty$ — equivalently, does $Z_\infty^\circ\subseteq\overline{X_\infty\setminus(\mathrm{Sing}\,X)_\infty}$ imply $Z_\infty^\circ\subseteq\overline{j^{-1}(\mathrm{Sing}\,X)\setminus(\mathrm{Sing}\,X)_\infty}$ ? This is known when $\dim\mathcal{O}_{X,\zeta}=1$ . (2) Transfer of arc-sharpness: for a proper birational morphism $f:X'\to X$ , under which conditions are the following equivalent: (i) $X$ is arc-sharp at $\zeta$ ; (ii) $X'$ is arc-sharp at each $\zeta'\in f^{-1}(\zeta)$ such that the field extension $k(\zeta')|k(\zeta)$ is separable? Implication (i) $\Rightarrow$ (ii) holds for proper surjective generically smooth morphisms, Theorem 5.5 gives the equivalence under the additional hypothesis that $f$ is finite birational and radicial over the punctured spectrum of $\mathcal{O}_{X,\zeta}$ , and Example 6.3 shows that some restriction relating $p$ and $\dim\mathcal{O}_{X,\zeta}$ (tentatively $p>\dim\mathcal{O}_{X,\zeta}$ ) is necessary; the authors note the equivalence may hold without further restriction on $f$ . A representative special case: for an irreducible threefold $X$ in characteristic $p\ge3$ , is $X_\infty$ irreducible if and only if for every resolution of singularities $\pi:Y\to X$ and every $\zeta\in X$ the fiber $\pi^{-1}(\zeta)$ has a point over the separable closure of $k(\zeta)$ ? (3) Curve selection at arc-blunt points: if $\zeta\in X$ and $f\in\mathcal{O}_X$ with $f\neq0$ , and $X$ is arc-blunt at $\zeta$ , does there exist a field extension $K\supseteq k(\zeta_\infty)$ and a wedge $\varphi:\mathrm{Spec}\,K[[u]]\to X_\infty$ (a one-parameter family of arcs) with $\varphi(0)=\zeta_\infty$ and $\varphi(\eta)\notin(V(f))_\infty$ , where $\eta$ is the generic point of $\mathrm{Spec}\,K[[u]]$ ? (4) Birational elimination of small components: for a valuation $v$ of the function field $k(X)|k$ with center $z\in X$ (the point whose local ring $\mathcal{O}_{X,z}$ is contained in the valuation ring of $v$ with compatible maximal ideals, i.e. the valuation ring dominates $\mathcal{O}_{X,z}$ ), does there exist a birational morphism $X_v\to X$ such that $v$ has a center on $X_v$ and the arc space $(X_v)_\infty$ is irreducible? In the related birational-equivalence form — is every irreducible proper $k$ -variety birationally equivalent to a proper variety $Y$ with $Y_\infty$ irreducible? — the answer is affirmative over algebraically closed fields for $p>d!$ with $d$ the dimension of the function field (Proposition 6.7), so the open content is small characteristics and the model-over- $X$ variant. The source tests the questions on the family of hypersurfaces $V(y_0^p+\sum_ib_iy_i^p)$ and its iterated variants.

References

Angélica Benito, Olivier Piltant, Ana J. Reguera. Small irreducible components of arc spaces in positive characteristic. Journal of Pure and Applied Algebra 226(11), 2022, article 107110. DOI: 10.1016/j.jpaa.2022.107110, https://doi.org/10.1016/j.jpaa.2022.107110 (HAL preprint hal-01945228, https://hal.science/hal-01945228).
Ana J. Reguera. Corrigendum: A curve selection lemma in spaces of arcs and the image of the Nash map. Compositio Mathematica 157(3), 2021, 641-648. DOI: 10.1112/s0010437x20007733, https://doi.org/10.1112/s0010437x20007733 (HAL hal-02423894, https://hal.science/hal-02423894).
Nguyen Hong Duc. Curve selection lemma in arc spaces. Preprint, 2022. arXiv:2301.00128, https://arxiv.org/abs/2301.00128.
Olivier Piltant, Ana J. Reguera. Algebraic properties of arc spaces. Preprint, 2024. HAL hal-04693205, https://hal.science/hal-04693205.
Christopher Chiu, Tommaso de Fernex, Roi Docampo. On arc fibers of morphisms of schemes. Journal of the European Mathematical Society 28(4), 2026, 1489-1531. DOI: 10.4171/JEMS/1500, https://doi.org/10.4171/JEMS/1500 (preprint arXiv:2206.08060, https://arxiv.org/abs/2206.08060).
Tommaso de Fernex, Shih-Hsin Wang. Arcs on rational double points in arbitrary characteristic. Preprint, 2025. arXiv:2508.12423, https://arxiv.org/abs/2508.12423.