In mathematics, the arithmetic genus of an algebraic variety is one of a few possible generalizations of the genus of an algebraic curve or Riemann surface...

graph genus problem is NP-complete. There are two related definitions of genus of any projective algebraic scheme X {\displaystyle X} : the arithmetic genus...

singular curve and the geometric genus of the desingularisation. The arithmetic genus is larger than the geometric genus, and the height of a point may...

geometry, the genus–degree formula relates the degree d {\displaystyle d} of an irreducible plane curve C {\displaystyle C} with its arithmetic genus g {\displaystyle...

duality thus implies that the arithmetic genus and the geometric genus coincide. They will simply be called the genus of X. Serre duality is also a key...

birational, the definition is extended by birational invariance. Genus (mathematics) Arithmetic genus Invariants of surfaces Danilov & Shokurov (1998), p. 53 P...

difference p g − p a {\displaystyle p_{g}-p_{a}} of the geometric genus and the arithmetic genus of more complicated surfaces. Surfaces are sometimes called...

topological genus, but, in dimension two, one needs to distinguish the arithmetic genus p a {\displaystyle p_{a}} and the geometric genus p g {\displaystyle...

group is finite can be replaced by the condition that it is not of arithmetic genus one and every non-singular rational component meets the other components...

suffices to show that the Todd genus agrees with the arithmetic genus for algebraic varieties as the arithmetic genus is also 1 for complex projective...

statement as above holds, provided that the geometric genus as defined above is replaced by the arithmetic genus ga, defined as g a := dim k ⁡ H 1 ( C , O C )...

characteristic of the trivial bundle, and is equal to 1 + pa, where pa is the arithmetic genus of the surface. For comparison, the Riemann–Roch theorem for a curve...

1969 when they proved that moduli spaces of stable curves of fixed arithmetic genus are proper smooth Deligne–Mumford stacks. If the "étale" is weakened...

{O}}_{X}\cong {\mathcal {O}}_{B}} and all fibers of f {\displaystyle f} have arithmetic genus g {\displaystyle g} . If X {\displaystyle X} is a smooth projective...

rational, because both are characterized by the vanishing of both the arithmetic genus and the second plurigenus. Zariski found some examples (Zariski surfaces)...

point. genus See #arithmetic genus, #geometric genus. genus formula The genus formula for a nodal curve in the projective plane says the genus of the...

measure. There are many ways to do this. For example, one can use the arithmetic genus of the curve. Noether's method takes a plane curve and repeatedly applies...

irreducible components of the nodal curve, the labelling of a vertex is the arithmetic genus of the corresponding component, edges correspond to nodes of the curve...

Masayoshi (1960), "On rational surfaces. I. Irreducible curves of arithmetic genus 0 or 1", Mem. Coll. Sci. Univ. Kyoto Ser. A Math., 32: 351–370, MR 0126443...

by Philip Wagreich in 1970, is a surface singularity such that the arithmetic genus of its local ring is 1. Rational singularity Wagreich, Philip (April...

mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is...

Complete intersection Serre duality Spaltenstein variety Arithmetic genus, geometric genus, irregularity Tangent space, Zariski tangent space Function...

between the symmetric algebra of a vector space and its dual. arithmetic genus The arithmetic genus of a variety is a variation of the Euler characteristic...

Grothendieck–Riemann–Roch theorem to arithmetic varieties. For this one defines arithmetic Chow groups CHp(X) of an arithmetic variety X, and defines Chern classes...

R(K_{X}):=\bigoplus _{d\geq 0}H^{0}(X,K_{X}^{d}).} Also see geometric genus and arithmetic genus. The Kodaira dimension of X is defined to be − ∞ {\displaystyle...

Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Part of the inspiration comes from complex...

( X , O X ) {\displaystyle H^{n}(X,{\mathcal {O}}_{X})} , and the arithmetic genus (according to one convention) is the alternating sum χ ( X , O X )...

q=h^{0,1}.} The geometric genus: p g = h 0 , 2 = h 2 , 0 = P 1 . {\displaystyle p_{g}=h^{0,2}=h^{2,0}=P_{1}.} The arithmetic genus: p a = p g − q = h 0 ,...

surfaces fibred over a curve where the general fibre is a curve of arithmetic genus one with a cusp. Once these adjustments are made, the surfaces are...

arithmetic groups. An arithmetic hyperbolic three-manifold is the quotient of hyperbolic space H 3 {\displaystyle \mathbb {H} ^{3}} by an arithmetic Kleinian...