The smooth surfaces in ℙ4 with few apparent triple points

2016 ◽  
Vol 18 (01) ◽  
pp. 1550013
Author(s):  
José Carlos Sierra
Keyword(s):  
Plane Curve ◽  
London Mathematical Society ◽  
Plane Curves ◽  
Smooth Surfaces ◽  
Projective Varieties ◽  
Cambridge University ◽  
Projective Character ◽  
Genus Formula ◽  
Triple Points ◽  
Complex Projective

We classify smooth complex projective surfaces in [Formula: see text] with [Formula: see text] apparent triple points, thus recovering and extending the results of Ascione [Sulle superficie immerse in un [Formula: see text], le cui trisecanti costituiscono complessi di [Formula: see text] ordine, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur.[Formula: see text]5[Formula: see text] 6 (1897) 162–169] and Severi [Intorno ai punti doppi impropri di una superficie generale dello spazio a quattro dimensioni, e a’ suoi punti tripli apparenti, Rend. Circ. Mat. Palermo 15 (1901) 33–51] for [Formula: see text], Marletta [Le superficie generali dell’ [Formula: see text] dotate di due punti tripli apparenti, Rend. Circ. Mat. Palermo 34 (1912) 179–186] for [Formula: see text], and Aure [The smooth surfaces in [Formula: see text] without apparent triple points, Duke Math. J. 57 (1988) 423–430] for [Formula: see text]. This is done thanks to a new projective character that can be introduced as a consequence of the main result of [K. Ranestad, On smooth plane curve fibrations in [Formula: see text], in Geometry of Complex Projective Varieties, Sem. Conf., Vol. 9 (Mediterranean, 1993), pp. 243–255; J. C. Sierra and A. L. Tironi, Some remarks on surfaces in [Formula: see text] containing a family of plane curves, J. Pure Appl. Algebra 209 (2007) 361–369; V. Beorchia and G. Sacchiero, Surfaces in [Formula: see text] with a family of plane curves, J. Pure Appl. Algebra 213 (2009) 1750–1755]. Going a bit further, we obtain some bounds on the Euler characteristic [Formula: see text] in terms of the degree [Formula: see text] and the sectional genus [Formula: see text] of a smooth surface in [Formula: see text]. For [Formula: see text], these bounds were first obtained in [A. B. Aure and K. Ranestad, The smooth surfaces of degree [Formula: see text] in [Formula: see text], in Complex Projective Geometry, London Mathematical Society Lecture Note Series, Vol. 179 (Cambridge University Press, Cambridge, 1992), pp. 32–46; K. Ranestad, On smooth surfaces of degree [Formula: see text] in the projective fourspace, Ph.D. thesis, Oslo (1988); S. Popescu, On smooth surfaces of degree [Formula: see text] in the projective fourspace, Dissertation, Saarbrücken (1993)]. Here we give a different argument based on liaison that works also for [Formula: see text] and that allows us to determine the triples [Formula: see text] of the smooth surfaces with [Formula: see text] apparent triple points.

scholarly journals On Plane Curves with Double and Triple Points

2016 ◽  
Vol 119 (1) ◽  
pp. 60 ◽  
Author(s):  
Nancy Abdallah
Keyword(s):  
Plane Curve ◽  
Plane Curves ◽  
Jacobian Ideal ◽  
Pole Order ◽  
Hodge Filtration ◽  
Triple Points

We describe in simple geometric terms the Hodge filtration on the cohomology $H^*(U)$ of the complement $U=\mathsf{P}^2 \setminus C$ of a plane curve $C$ with ordinary double and triple points. Relations to Milnor algebra, syzygies of the Jacobian ideal and pole order filtration on $H^2(U)$ are given.

scholarly journals ALEXANDER POLYNOMIALS OF COMPLEX PROJECTIVE PLANE CURVES

2018 ◽  
Vol 97 (3) ◽  
pp. 386-395 ◽  
Author(s):  
QUY THUONG LÊ
Keyword(s):  
Projective Plane ◽  
Plane Curve ◽  
Alexander Polynomial ◽  
Plane Curves ◽  
Complex Projective Plane ◽  
Multiplier Ideals ◽  
Alexander Polynomials ◽  
Irreducible Components ◽  
Complex Projective

We compute the Alexander polynomial of a nonreduced nonirreducible complex projective plane curve with mutually coprime orders of vanishing along its irreducible components in terms of certain multiplier ideals.

scholarly journals On 3-syzygy and unexpected plane curves

2021 ◽  
Author(s):  
Grzegorz Malara ◽  
Piotr Pokora ◽  
Halszka Tutaj-Gasińska
Keyword(s):  
Projective Plane ◽  
Plane Curves ◽  
Complex Projective Plane ◽  
Geometric Properties ◽  
Complex Projective

AbstractIn this note we study curves (arrangements) in the complex projective plane which can be considered as generalizations of free curves. We construct families of arrangements which are nearly free and possess interesting geometric properties. More generally, we study 3-syzygy curve arrangements and we present examples that admit unexpected curves.

scholarly journals Connected Numbers and the Embedded Topology of Plane Curves

2018 ◽  
Vol 61 (3) ◽  
pp. 650-658 ◽  
Author(s):  
Taketo Shirane
Keyword(s):  
Plane Curve ◽  
Smooth Curve ◽  
Plane Curves ◽  
Irreducible Curve ◽  
Galois Cover ◽  
Splitting Number

AbstractThe splitting number of a plane irreducible curve for a Galois cover is effective in distinguishing the embedded topology of plane curves. In this paper, we define the connected number of a plane curve (possibly reducible) for a Galois cover, which is similar to the splitting number. By using the connected number, we distinguish the embedded topology of Artal arrangements of degree b ≥ 4, where an Artal arrangement of degree b is a plane curve consisting of one smooth curve of degree b and three of its total inflectional tangents.

scholarly journals Plane Curves Which are Quantum Homogeneous Spaces

2021 ◽  
Author(s):  
Ken Brown ◽  
Angela Ankomaah Tabiri
Keyword(s):  
Hopf Algebras ◽  
Final Section ◽  
Homogeneous Spaces ◽  
Plane Curve ◽  
Plane Curves ◽  
Closed Field ◽  
Quantum Homogeneous Space ◽  
Central Elements ◽  
Pointed Hopf Algebras ◽  
Future Work

AbstractLet $\mathcal {C}$ C be a decomposable plane curve over an algebraically closed field k of characteristic 0. That is, $\mathcal {C}$ C is defined in k2 by an equation of the form g(x) = f(y), where g and f are polynomials of degree at least two. We use this data to construct three affine pointed Hopf algebras, A(x, a, g), A(y, b, f) and A(g, f), in the first two of which g [resp. f ] are skew primitive central elements, with the third being a factor of the tensor product of the first two. We conjecture that A(g, f) contains the coordinate ring $\mathcal {O}(\mathcal {C})$ O ( C ) of $\mathcal {C}$ C as a quantum homogeneous space, and prove this when each of g and f has degree at most five or is a power of the variable. We obtain many properties of these Hopf algebras, and show that, for small degrees, they are related to previously known algebras. For example, when g has degree three A(x, a, g) is a PBW deformation of the localisation at powers of a generator of the downup algebra A(− 1,− 1,0). The final section of the paper lists some questions for future work.

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