In classical
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, the genus–degree formula relates the degree
of an
irreducible plane curve
In mathematics, a plane curve is a curve in a plane that may be a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane c ...
with its
arithmetic genus via the formula:
:
Here "plane curve" means that
is a closed curve in the
projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, paral ...
. If the curve is non-singular the
geometric genus
In algebraic geometry, the geometric genus is a basic birational invariant of algebraic varieties and complex manifolds.
Definition
The geometric genus can be defined for non-singular complex projective varieties and more generally for complex ...
and the
arithmetic genus are equal, but if the curve is singular, with only ordinary singularities, the geometric genus is smaller. More precisely, an ordinary
singularity of multiplicity
decreases the genus by
.
Motivation
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...
s are parametrized by
Weierstrass elliptic function
In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions is also referred to as ℘-functions and they are usually denoted by the s ...
s. Since these parameterizing functions are doubly periodic, the elliptic curve can be identified with a period parallelogram with the sides glued together ''i.e.'' a torus. So the genus of an elliptic curve is 1. The genus–degree formula is a generalization of this fact to higher genus curves. The basic idea would be to use higher degree equations. Consider the quartic equation
For small nonzero
this is gives the nonsingular curve. However, when
, this is
a reducible curve (the union of a nonsingular cubic and a line). When the points of infinity are added, we get a line meeting the cubic in 3 points. The complex picture of this reducible curve looks like a torus and a sphere touching at 3 points. As
changes to nonzero values, the points of contact open up into tubes connecting the torus and sphere, adding 2 handles to the torus, resulting in a genus 3 curve.
In general, if
is the genus of a curve of degree
nonsingular curve, then proceeding as above, we obtain a nonsingular curve of degree
by
-smoothing the union of a curve of degree
and a line. The line meets the degree
curve in
points, so this leads to an recursion relation
This recursion relation has the solution
.
Proof
The genus–degree formula can be proven from the
adjunction formula
In mathematics, especially in algebraic geometry and the theory of complex manifolds, the adjunction formula relates the canonical bundle of a variety and a hypersurface inside that variety. It is often used to deduce facts about varieties embedde ...
; for details, see .
[''Algebraic geometry'', ]Robin Hartshorne
__NOTOC__
Robin Cope Hartshorne ( ; born March 15, 1938) is an American mathematician who is known for his work in algebraic geometry.
Career
Hartshorne was a Putnam Fellow in Fall 1958 while he was an undergraduate at Harvard University (under ...
, Springer GTM 52, , chapter V, example 1.5.1
Generalization
For a non-singular
hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidea ...
of degree
in the
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
of
arithmetic genus the formula becomes:
:
where
is the
binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
.
Notes
See also
*
Thom conjecture
References
*
*
Enrico Arbarello, Maurizio Cornalba,
Phillip Griffiths,
Joe Harris. Geometry of algebraic curves. vol 1 Springer, , appendix A.
*
Phillip Griffiths and
Joe Harris, Principles of algebraic geometry, Wiley, , chapter 2, section 1.
*
Robin Hartshorne
__NOTOC__
Robin Cope Hartshorne ( ; born March 15, 1938) is an American mathematician who is known for his work in algebraic geometry.
Career
Hartshorne was a Putnam Fellow in Fall 1958 while he was an undergraduate at Harvard University (under ...
(1977): ''Algebraic geometry'', Springer, .
*
{{DEFAULTSORT:Genus-degree formula
Algebraic curves