scholarly journals A polynomial invariant for plane curve complements: Krammer polynomials

2018 ◽  
Author(s):  
Mehmet Aktas ◽  
Serdar Cellat ◽  
Hubeyb Gurdogan
Keyword(s):  
Plane Curve ◽  
Polynomial Invariant

scholarly journals Higher rank Clifford indices of curves on a K3 surface

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Soheyla Feyzbakhsh ◽  
Chunyi Li
Keyword(s):  
Vector Bundles ◽  
Plane Curve ◽  
Smooth Curve ◽  
Line Bundles ◽  
K3 Surface ◽  
Clifford Index ◽  
Smooth Plane ◽  
Higher Rank ◽  
Rank Vector ◽  
Semistable Vector Bundle

AbstractLet (X, H) be a polarized K3 surface with $$\mathrm {Pic}(X) = \mathbb {Z}H$$ Pic ( X ) = Z H , and let $$C\in |H|$$ C ∈ | H | be a smooth curve of genus g. We give an upper bound on the dimension of global sections of a semistable vector bundle on C. This allows us to compute the higher rank Clifford indices of C with high genus. In particular, when $$g\ge r^2\ge 4$$ g ≥ r 2 ≥ 4 , the rank r Clifford index of C can be computed by the restriction of Lazarsfeld–Mukai bundles on X corresponding to line bundles on the curve C. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher rank vector bundles. We also apply the same method to the projective plane and show that the rank r Clifford index of a degree $$d(\ge 5)$$ d ( ≥ 5 ) smooth plane curve is $$d-4$$ d - 4 , which is the same as the Clifford index of the curve.

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 FROM RIBBON CATEGORIES TO GENERALIZED YANG–BAXTER OPERATORS AND LINK INVARIANTS (AFTER KITAEV AND WANG)

2013 ◽  
Vol 24 (01) ◽  
pp. 1250126 ◽  
Author(s):  
SEUNG-MOON HONG
Keyword(s):  
The Other ◽  
Polynomial Invariant ◽  
Link Invariants ◽  
Fusion Categories ◽  
New Family ◽  
Kauffman Polynomial ◽  
Twist Structure

We consider two approaches to isotopy invariants of oriented links: one from ribbon categories and the other from generalized Yang–Baxter (gYB) operators with appropriate enhancements. The gYB-operators we consider are obtained from so-called gYBE objects following a procedure of Kitaev and Wang. We show that the enhancement of these gYB-operators is canonically related to the twist structure in ribbon categories from which the operators are produced. If a gYB-operator is obtained from a ribbon category, it is reasonable to expect that two approaches would result in the same invariant. We prove that indeed the two link invariants are the same after normalizations. As examples, we study a new family of gYB-operators which is obtained from the ribbon fusion categories SO (N)2, where N is an odd integer. These operators are given by 8 × 8 matrices with the parameter N and the link invariants are specializations of the two-variable Kauffman polynomial invariant F.

A Polynomial Invariant of Graphs On Orientable Surfaces

2001 ◽  
Vol 83 (3) ◽  
pp. 513-531 ◽  
Author(s):  
Béla Bollobás ◽  
Oliver Riordan
Keyword(s):  
Polynomial Invariant ◽  
Orientable Surfaces

scholarly journals On the approximate Jacobian Newton diagrams of an irreducible plane curve

2013 ◽  
Vol 65 (1) ◽  
pp. 169-182
Author(s):  
Evelia Rosa GARCÍA BARROSO ◽  
Janusz GWOŹDZIEWICZ
Keyword(s):  
Plane Curve ◽  
Approximate Jacobian

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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