Del Pezzo surfaces, rigid line configurations and Hirzebruch–Kummer coverings

Bauer, Ingrid; Catanese, Fabrizio

doi:10.1007/s40574-018-0169-x

Cited by 2 publications

(8 citation statements)

References 15 publications

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“…The orbit of s 21 inside M ′ for the subgroup generated by 3 and ∶= (1, 2, 3)(4, 5) has at least 7 elements, hence it equals M ′ . Since (3,4) sends s 21 to − 32 , follows that M is a single 5 -orbit. Hence the stabilizer of s 21 has cardinality 5: but we know that it contains (1,2,3,4,5).…”

Section: Remark 32mentioning

confidence: 98%

“…Proof The bijection follows immediately from the fact that we have two transitive actions, and the stabilizer of s 21 is the cyclic subgroup generated by (1,2,3,4,5), which is also the stabilizer of the standard pentagon corresponding to the identity map. We define then (m) by the property that it associates to an oriented pentagon p(i) the neighbouring oriented pentagon p(2i); from this definition follows that 2 4 ◻…”

Section: Definingmentioning

confidence: 98%

“…Del Pezzo surfaces are projectively unique as soon as n ≥ 5 , and we are interested (see [3], and the related papers [2] and [1]), about their defining equations, here particularly in the case n = 5.…”

Section: Introductionmentioning

confidence: 99%

“…Finally, in a previous paper [3] we have written explicit equations of Del Pezzo surfaces in products (ℙ 1 ) h , and here, for the Del Pezzo of degree 5, we give in Theorem 5.3 5 -invariant equations inside (ℙ 1 ) 5 .…”

Section: Introductionmentioning

confidence: 99%

“…We can summarize part of our discussion in the following corollary. 3 Corollary 3.4 There is a natural bijection between the set M and the set of oriented nondegenerate combinatorial pentagons on {1, 2, 3, 4, 5} , in such a way that −m is the oppositely oriented pentagon of m. M∕ ± 1 is then in bijection with the set P of pentagons, where ± ij corresponds to the neighbouring pentagon of ±s ij .…”

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confidence: 99%

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$${\mathfrak {S}}_5$$-equivariant syzygies for the Del Pezzo surface of degree 5

Bauer

Catanese

2020

Rend. Circ. Mat. Palermo, II. Ser

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The Del Pezzo surface Y of degree 5 is the blow up of the plane in 4 general points, embedded in $${\mathbb {P}}^5$$P5 by the system of cubics passing through these points. It is the simplest example of the Buchsbaum–Eisenbud theorem on arithmetically-Gorenstein subvarieties of codimension 3 being Pfaffian. Its automorphism group is the symmetric group $${\mathfrak {S}}_5$$S5. We give canonical explicit $${\mathfrak {S}}_5$$S5-invariant Pfaffian equations through a 6$$\times $$×6 antisymmetric matrix. We give concrete geometric descriptions of the irreducible representations of $${\mathfrak {S}}_5$$S5. Finally, we give $${\mathfrak {S}}_5$$S5-invariant equations for the embedding of Y inside $$({\mathbb {P}}^1)^5$$(P1)5, and show that they have the same Hilbert resolution as for the Del Pezzo of degree 4.

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