Reduction type of smooth plane quartics

AbstractWe study new families of curves that are suitable for efficiently parametrizing their moduli spaces. We explicitly construct such families for smooth plane quartics in order to determine unique representatives for the isomorphism classes of smooth plane quartics over finite fields. In this way, we can visualize the distributions of their traces of Frobenius. This leads to new observations on fluctuations with respect to the limiting symmetry imposed by the theory of Katz and Sarnak.

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The Characteristic Numbers of Quartic Plane Curves

Canadian Journal of Mathematics ◽

10.4153/cjm-1999-048-1 ◽

1999 ◽

Vol 51 (5) ◽

pp. 1089-1120 ◽

Cited By ~ 5

Author(s):

Ravi Vakil

Keyword(s):

Moduli Space ◽

Intersection Theory ◽

Plane Curves ◽

Stable Maps ◽

Characteristic Numbers ◽

Plane Quartics ◽

Smooth Plane

AbstractThe characteristic numbers of smooth plane quartics are computed using intersection theory on a component of the moduli space of stable maps. This completes the verification of Zeuthen’s prediction of characteristic numbers of smooth plane curves. A short sketch of a computation of the characteristic numbers of plane cubics is also given as an illustration.

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Normal forms of smooth plane quartics and their restrictions

ScienceAsia ◽

10.2306/scienceasia1513-1874.2016.42s.026 ◽

2016 ◽

Vol 42S (1) ◽

pp. 26

Author(s):

Takahashi Tadashi

Keyword(s):

Normal Forms ◽

Plane Quartics ◽

Smooth Plane

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Higher rank Clifford indices of curves on a K3 surface

Selecta Mathematica ◽

10.1007/s00029-021-00664-z ◽

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.

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Diffraction of plane elastic waves by a crack, with application to a problem of brittle fracture

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700028354 ◽

1963 ◽

Vol 3 (3) ◽

pp. 325-339 ◽

Cited By ~ 2

Author(s):

M. Papadopoulos

Keyword(s):

Elastic Waves ◽

Scattered Field ◽

Velocity Potential ◽

Elastic Solid ◽

Step Function ◽

Free Surface Waves ◽

Dependent Variables ◽

Smooth Plane ◽

New Feature ◽

Set Up

AbstractA crack is assumed to be the union of two smooth plane surfaces of which various parts may be in contact, while the remainder will not. Such a crack in an isotropic elastic solid is an obstacle to the propagation of plane pulses of the scalar and vector velocity potential so that both reflected and diffracted fields will be set up. In spite of the non-linearity which is present because the state of the crack, and hence the conditions to be applied at the surfaces, is a function of the dependent variables, it is possible to separate incident step-function pulses into either those of a tensile or a compressive nature and the associated scattered field may then be calculated. One new feature which arises is that following the arrival of a tensile field which tends to open up the crack there is necessarily a scattered field which causes the crack to close itself with the velocity of free surface waves.

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