The $2$-Hessian and sextactic points on plane algebraic curves

Maugesten, Paul Aleksander; Moe, Torgunn Karoline

doi:10.7146/math.scand.a-114715

Cited by 5 publications

(4 citation statements)

References 13 publications

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“…In [2] Cayley defined the second Hessian of a curve and proved that the sextactic points are as the flexes, the common points of the curve and its Hessian. The formula that Cayley provided was corrected in [9]. We have the following definition.…”

Section: Preliminariesmentioning

confidence: 99%

Construction of Free Curves by Adding Lines to a Given Curve

Dimca,

Ilardi,

Pokora

et al. 2023

Results Math

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In the present note we construct new families of free plane curves starting from a curve C and adding high order inflectional tangent lines of C, lines joining the singularities of the curve C, or lines in the tangent cone of some singularities of C. These lines L have in common that the intersection $$C \cap L$$ C ∩ L consists of a small number of points. We introduce the notion of a supersolvable plane curve and conjecture that such curves are always free, as in the known case of line arrangements. Some evidence for this conjecture is given as well, both in terms of a general result in the case of quasi homogeneous singularities and in terms of specific examples. We construct a new example of maximizing curve in degree 8 and the first and unique known example of maximizing curve in degree 9. In the final section, we use a stronger version of a result due to Schenck, Terao and Yoshinaga to construct families of free conic-line arrangements by adding lines to the conic-line arrangements of maximal Tjurina number recently classified by Beorchia and Miró-Roig in arXiv:2303.04665

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Section: Preliminariesmentioning

confidence: 99%

Construction of Free Curves by Adding Lines to a Given Curve

Dimca,

Ilardi,

Pokora

et al. 2023

Results Math

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“…In order to answer this question Cayley introduced the notion of the second Hessian of a curve and proved that the sextactic points are, in analogy to the flexes, the common points of the curve and its second Hessian, which defines a curve of degree 12 deg(Γ) − 27. Only recently Maugesten and Moe [21] checked Cayley's formula carefully and pinned down an inaccuracy in its coefficients. Since the erroneous formula was repeated in the literature for over 150 years, we state it here in the correct form following the notation of [21].…”

Section: Introductionmentioning

confidence: 99%

“…Only recently Maugesten and Moe [21] checked Cayley's formula carefully and pinned down an inaccuracy in its coefficients. Since the erroneous formula was repeated in the literature for over 150 years, we state it here in the correct form following the notation of [21]. The inaccuracy of Cayley was to write 40 in the place of the coefficient 20 appearing in the third line of the formula in Definition 3.1.…”

Section: Introductionmentioning

confidence: 99%

Sextactic points on the Fermat cubic curve and arrangements of conics

Szemberg¹,

Szpond²

2022

Preprint

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The purpose of this note is to report, in narrative rather than rigorous style, about the nice geometry of 6-division points on the Fermat cubic F and various conics naturally attached to them. Most facts presented here were derived by symbolic algebra programs and the idea of the note is to propose a research direction for searching for conceptual proofs of facts stated here and their generalisations. Extensions in several directions seem possible (taking curves of higher degree and contact to F , studying higher degree curves passing through higher order division points on F , studying curves passing through intersection points of already constructed curves, taking the duals etc.) and we hope some younger colleagues might find pleasure in following proposed paths as well as finding their own.

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“…A similar argument can be used to prove that in a 1-parameter family of curves, hyperflexes are points of multiplicity 2 in the divisor of flexes, and flexes are points of multiplicity 1 in the divisor of sextactic points-points at which the osculating conic meets the curve with intersection multiplicity at least 6. For more on the enumerative geometry of sextactic points, refer to [Cay09, § 341] and[MM17].17 We say "expected" because of the assumption that the linear system of plane conics is general in the sense of Remark 16.…”

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confidence: 99%

Inflectionary Invariants for Isolated Complete Intersection Curve Singularities

Patel¹,

Swaminathan²

2017

Preprint

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We investigate the role played by curve singularity germs in the enumeration of inflection points in families of curves acquiring singular members. Let N ≥ 2, and consider an isolated complete intersection curve singularity germ f : (C N , 0) → (C N−1 , 0). We introduce a numerical function m → AD m(2) (f) that arises as an error term when counting m th -order weight-2 inflection points with ramification sequence (0, . . . , 0, 2) in a 1parameter family of curves acquiring the singularity f = 0, and we compute AD m(2) (f) for various (f, m). Particularly, for a node defined by f : (x, y) → xy, we prove that AD m(2) (xy) = m+1 4, and we deduce as a corollary that AD mfor any f, where mult 0 ∆ f is the multiplicity of the discriminant ∆ f at the origin in the deformation space. Furthermore, we show that the function m → AD m(2is an analytic invariant measuring how much the singularity "counts as" an inflection point. We obtain similar results for weight-2 inflection points with ramification sequence (0, . . . , 0, 1, 1) and for weight-1 inflection points, and we apply our results to solve various related enumerative problems. CONTENTSANAND P. PATEL AND ASHVIN A. SWAMINATHAN 6.3. Computation of AD 2 (1,1) (f) 50 6.4. Computation of AD 3 (2) (y t − x s ) 51 6.5. Computation of AD 2 (1) (y t − x s ) 55 6.6. Computation of AD 4 (2) (y 2 − x s ) 59 6.7. Computation of AD 3 (1) (y 2 − x s ) 62 6.8.

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The $2$-Hessian and sextactic points on plane algebraic curves

Cited by 5 publications

References 13 publications

Construction of Free Curves by Adding Lines to a Given Curve

Construction of Free Curves by Adding Lines to a Given Curve

Sextactic points on the Fermat cubic curve and arrangements of conics

Inflectionary Invariants for Isolated Complete Intersection Curve Singularities

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