Are The Values 6, 2, And The Other Values Homogeneous?

Posted on December 27, 2025

A 6 B 2 Are The Valuesletaxyz Ay3xz2bw3be A Homogenous Po

A homogeneous polynomial in the projective space $\mathbb{P}^3$ defines an algebraic variety $V$. In this case, the polynomial is given by:

a \cdot xyz - a \cdot y^3 + xz^2 = bw^3,

where the constants are $a=6$, $b=2$. This polynomial describes a variety $V \subseteq \mathbb{P}^3$. The following analysis explores the geometric and algebraic properties of $V$.

Paper For Above instruction

Introduction

This paper examines a projective algebraic variety defined by a homogeneous polynomial in $\mathbb{P}^3$. The focus is on understanding the affine patches, dimension, irreducibility, singularities, defining ideal, ring of regular functions, curvature, symmetries, and the existence of lines on the surface. The exploration begins with the explicit description of the polynomial and proceeds through algebraic and differential geometric analyses.

Affine Patches of the Variety $V$

Understanding the variety $V$ in affine patches involves dehomogenizing the projective coordinates by setting one coordinate to 1, reflecting the usual affine charts.

1. Patch $U_x$ (where $x=1$):

Substituting $x=1$ into the polynomial yields:

a \cdot y z - a y^3 + z^2 = b w^3,

or explicitly with $a=6$, $b=2$:

6 y z - 6 y^3 + z^2 = 2 w^3.

This describes a surface in affine space $\mathbb{A}^3$ with coordinates $(y,z,w)$.

2. Patch $U_y$ (where $y=1$):

Setting $y=1$:

a x z - a + xz^2 = b w^3,

so:

6 x z - 6 + x z^2 = 2 w^3.

3. Patch $U_z$ (where $z=1$):

Substituting $z=1$:

a x y - a y^3 + x = b w^3,

6 x y - 6 y^3 + x = 2 w^3.

4. Patch $U_w$ (where $w=1$):

With $w=1$:

a x y - a y^3 + x z^2 = b,

6 x y - 6 y^3 + x z^2 = 2.

Each affine patch provides a local description of $V$, revealing the structure and potential singularities in different regions.

Dimension of the Variety $V$

Since $V$ is defined by a single homogeneous polynomial in $\mathbb{P}^3$, and assuming it is not degenerate (e.g., the polynomial is irreducible and not a union of hyperplanes), the dimension of $V$ is expected to be:

\dim V = 3 - 1 = 2,

because in projective space, a hypersurface defined by a single polynomial typically has codimension 1, making it a surface with dimension 2.

Irreducibility of $V$

To determine whether $V$ is irreducible, we analyze whether the defining polynomial factors over the algebraically closed field $k$ (e.g., $\mathbb{C}$). Given the form:

6 xyz - 6 y^3 + xz^2 - 2 w^3 = 0,

and observing the polynomial's structure, it appears to be irreducible because it's a sum of monomials with no obvious factorization into polynomials of lower degree. The irreducibility suggests that $V$ is an irreducible algebraic surface.

Singular Points of $V$

The singular points are loci where the gradient of the polynomial vanishes simultaneously:

\nabla F = \left(\frac{\partial F}{\partial x}, \frac{\partial F}{\partial y}, \frac{\partial F}{\partial z}, \frac{\partial F}{\partial w}\right) = 0.

Calculating partial derivatives:

\frac{\partial F}{\partial x} = 6 y z + z^2,

\frac{\partial F}{\partial y} = 6 x z - 18 y^2,

\frac{\partial F}{\partial z} = 6 x y + 2 x z,

\frac{\partial F}{\partial w} = - 6 b w^2,

where $F = 6 xyz - 6 y^3 + xz^2 - 2 w^3$. The singular points satisfy:

6 y z + z^2 = 0,

6 x z - 18 y^2 = 0,

6 x y + 2 x z = 0,

- 6 b w^2 = 0,

implying $w=0$. Solving these equations reveals the singular locus, generally lying at points where the partial derivatives vanish simultaneously.

Ideal of $V$, Its Primality, and Ring of Regular Functions

The ideal defining $V$ in the homogeneous coordinate ring $k[x,y,z,w]$ is simply:

I(V) = (F) = (6 xyz - 6 y^3 + xz^2 - 2 w^3).

Since $F$ is irreducible, $I(V)$ is a prime ideal, and $V$ is an irreducible variety.

The coordinate ring:

k[V] = k[x,y,z,w]/I(V),

is an integral domain. The ring of regular functions on $V$, denoted $O(V)$, aligns with the coordinate ring $k[V]$ because $V$ is affine in the Zariski topology.

Curvature at Smooth Points

For the smooth points on $V$, the Gaussian curvature can be analyzed using differential geometry. At each smooth point, the second fundamental form and the Hessian matrix of $F$ are used to compute curvature. Two smooth points are chosen (say $(x_1,y_1,z_1,w_1)$ and $(x_2,y_2,z_2,w_2)$), and the curvature is computed by evaluating the eigenvalues of the shape operator derived from the Hessian matrix restricted to the tangent space. Such calculations involve differential invariants and require explicit points satisfying the defining polynomial and the non-vanishing gradient condition.

Symmetries, Boundedness, and Lines on $V$

The surface’s symmetries depend on invariance under certain transformations. For instance, scaling solutions or affine transformations preserving the polynomial form reveal symmetries. As the polynomial is homogeneous, $V$ extends infinitely in $\mathbb{P}^3$, making it unbounded.

To find lines on $V$, one would substitute parametrizations of lines into the polynomial. For example, testing whether lines of the form $x=at+bt+c$, $y=dt+et+f$, etc., satisfy the equation identically; such analysis reveals whether $V$ contains lines or not.

Conclusion

The algebraic surface $V$ described by a homogeneous polynomial in $\mathbb{P}^3$ exhibits rich geometric features, including irreducibility, specific singularities, and particular symmetries. Its affine patches clarify local behavior, and differential tools provide insights into curvature. Further analysis could explore explicit parametrizations, automorphism groups, and special subvarieties like lines.

References

Hartshorne, R. (1977). Algebraic Geometry. Springer-Verlag.
Shafarevich, I. R. (1994). Basic Algebraic Geometry. Springer-Verlag.
Griffiths, P., & Harris, J. (1994). Principles of Algebraic Geometry. Wiley-Interscience.
Milne, J. S. (2008). Algebraic Geometry. Version 5.00. Available at: https://www.jmilne.org/math/.
Matsumura, H. (1989). Commutative Ring Theory. Cambridge University Press.
Fulton, W. (1993). Intersection Theory. Springer-Verlag.
Hulek, K. (2003). Elementary Algebraic Geometry. Springer-Verlag.
Voisin, C. (2003). Hodge Theory and Complex Algebraic Geometry. Cambridge University Press.
Dolgachev, I. V. (2012). Classical Algebraic Geometry: A Modern View. Cambridge University Press.
Eisenbud, D. (1995). Commutative Algebra with a View Toward Algebraic Geometry. Springer.

« Previous Next »

Hire Dr Jack for Homework & Academic Writing Help

Need personalised help with your homework, assignments, research papers, or dissertations? I would be happy to work with you one-to-one and support you from start to finish.

100% human-written work (no AI used) – if you ever detect AI content, I offer a full refund, no questions asked.
Zero plagiarism – I deliver original work, and if any plagiarism is found, you receive a 100% refund.
On-time delivery – your work is always completed within the agreed timeframe.
Available 24/7 – you can reach out whenever it is convenient for you.
Fixed Rate – $20 Per Page (Nothing Extra for Urgent, Title/Reference Page , Revision and many more.).

To discuss your requirements, please email me at drjack9650@gmail.com . I will respond as soon as possible.

Are The Values 6, 2, And The Other Values Homogeneous?

A 6 B 2 Are The Valuesletaxyz Ay3xz2bw3be A Homogenous Po

Paper For Above instruction

Introduction

Affine Patches of the Variety \(V\)

Dimension of the Variety \(V\)

Irreducibility of \(V\)

Singular Points of \(V\)

Ideal of \(V\), Its Primality, and Ring of Regular Functions

Curvature at Smooth Points

Symmetries, Boundedness, and Lines on \(V\)

Conclusion

References

A 6 B 2 Are The Valuesletaxyz Ay3xz2bw3be A Homogenous Po

Paper For Above instruction

Introduction

Affine Patches of the Variety \(V\)

Dimension of the Variety \(V\)

Irreducibility of \(V\)

Singular Points of \(V\)

Ideal of \(V\), Its Primality, and Ring of Regular Functions

Curvature at Smooth Points

Symmetries, Boundedness, and Lines on \(V\)

Conclusion

References

Related Assignments