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Narváez Macarro, Luis; Reguera, Ana J.; Tirado Hernández, María de la Paz

Finiteness of leaps in the sense of Hasse-Schmidt of unibranch curves in positive characteristic. (English) Zbl 1537.13007

Adv. Math. 437, Article ID 109441, 19 p. (2024).
According to the abstract, the authors prove “that the set of leaps of the chain of \(m\)-integrable derivations of a curve \(X\) over a perfect field with geometrically unibranch singularities is finite. This result is a consequence of an affirmative answer to Seidenberg’s question of extending, in positive characteristic, Hasse-Schmidt derivations of finite length of the local rings of \(X\) to their integral closures.” The last section of the paper is devoted to some questions about the relationship between leaps and other positive characteristic invariants, as F-threshold or more generally F-jumping numbers. They mention – and give appropriate references for the proofs of the statements – that F-jumping numbers do not change when taking the integral closure of an ideal, but leaps may do. In the case of unibranch curves, they also ask for the relationship of the set of leaps with the ramification of the generic plane projection, or with invariants associated with the motivic series in positive characteristic. They also say that a main obstacle in the theory is the lacking of an algorithm computing the leaps of a given commutative (computable) algebra. The last sentence of the paper is that the “theory of leaps” is still in a very early stage.
Reviewer: Francisco Castro-Jiménez (Sevilla)

MSC:

13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
13N15 Derivations and commutative rings
14B05 Singularities in algebraic geometry
14H20 Singularities of curves, local rings

Keywords:

Hasse-Schmidt derivation; integrable derivation; leaps; arcs space; unibrach curves; integral closure
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References:

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