Automorphisms and the Canonical Ideal

Kontogeorgis, Aristides; Terezakis, Alexios; Tsouknidas, Ioannis

doi:10.1007/s00009-021-01878-3

Cited by 3 publications

(2 citation statements)

References 13 publications

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“…The reader may observe that this decomposition of H 0 (F n , Ω ⊗m Fn ) is closely related to the classic result of M. Noether, F. Enriques and K. Petri on the canonical ideal of non-hyperelliptic curves: indeed the graded ring W m is isomorphic to the quotient of Sym H 0 (F n , Ω Fn ) by some binomial relations, whereas the elements of I m are the missing generators for the kernel of the canonical map Sym H 0 (F n , Ω Fn ) ։ H 0 (F n , Ω ⊗m Fn ). For more details on the explicit construction in the case of Fermat curves see [11], an application of a more general technique given in [2]. We proceed with the description of the action of G on W 1 = H 0 (F n , Ω Fn ):…”

Section: Character Tables For M-differentialsmentioning

confidence: 99%

The equivariant Hilbert series of the canonical ring of Fermat curves

Charalambous¹,

Karagiannis²,

Karanikolopoulos³

et al. 2021

Preprint

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We consider a Fermat curve Fn : x n + y n + z n = 1 over an algebraically closed field k of characteristic p ≥ 0 and study the action of the automorphism group G = (Z/nZ × Z/nZ) ⋊ S 3 on the canonical ring R = H 0 (Fn, Ω ⊗m Fn ) when p > 3, p ∤ n and n − 1 is not a power of p. In particular, we explicitly determine the classes [H 0 (Fn, Ω ⊗mFn )] in the Grothendieck group K 0 (G, k) of finitely generated k[G]-modules, describe the respective equivariant Hilbert series H R,G (t) as a rational function, and use our results to write a program in Sage that computes H R,G (t) for an arbitrary Fermat curve.

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Section: Character Tables For M-differentialsmentioning

confidence: 99%

The equivariant Hilbert series of the canonical ring of Fermat curves

Charalambous¹,

Karagiannis²,

Karanikolopoulos³

et al. 2021

Preprint

Self Cite

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“…The HKG-covers proved to be an important tool in the study of local actions and the deformation theory of curves with automorphisms, see e.g. [3], [6], [7] [26], [27], [28] and [20]. For any Q ∈ Y (k), one may construct an HKG-cover X Q → P 1 that approximates the cover π : X → Y locally over Q, see below for a precise definition.…”

Section: Introductionmentioning

confidence: 99%

𝑝-group Galois covers of curves in characteristic 𝑝

Garnek

2023

Trans. Amer. Math. Soc.

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We study cohomologies of a curve with an action of a finite p p -group over a field of characteristic p p . Assuming the existence of a certain “magical element” in the function field of the curve, we compute the equivariant structure of the module of holomorphic differentials and the de Rham cohomology, up to certain local terms. We show that a generic p p -group cover has a “magical element”. As an application we compute the de Rham cohomology of a curve with an action of a finite cyclic group of prime order.

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Representations on canonical models of generalized fermat curves and their syzygies

Karagiannis

2024

The Quarterly Journal of Mathematics

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We study canonical models of $\left(\mathbb{Z}/k\mathbb{Z}\right)^n$-covers of the projective line, tamely ramified at exactly n + 1 points each of index k, when $k,n\geq 2$ and the characteristic of the ground field K is either zero or does not divide k. We determine explicitly the structure of the respective homogeneous coordinate ring first as a graded K-algebra, next as a $\left(\mathbb{Z}/k\mathbb{Z}\right)^n$- representation over K and then as a graded module over the polynomial ring; in the latter case, we give generators for its first syzygy module, which we also decompose as a direct sum of irreducible representations.

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Automorphisms and the Canonical Ideal

Cited by 3 publications

References 13 publications

The equivariant Hilbert series of the canonical ring of Fermat curves

The equivariant Hilbert series of the canonical ring of Fermat curves

𝑝-group Galois covers of curves in characteristic 𝑝

Representations on canonical models of generalized fermat curves and their syzygies

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