11.2 Algebraic Geometry Problems

Gröbner bases are powerful tools in algebraic geometry. They help solve polynomial equations, find dimensions of varieties, and determine irreducible components. These techniques are crucial for understanding the structure of algebraic varieties.

Algebraic geometry problems often involve analyzing varieties' dimensions, degrees, and intersections. By using Gröbner bases, we can tackle these complex issues and gain insights into the fundamental properties of algebraic objects.

Algebraic Geometry Problems

Applications of Gröbner bases

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• Solve systems of polynomial equations by computing Gröbner bases and using the elimination property (Buchberger's algorithm)
• Determine dimension of algebraic varieties equal to number of variables minus number of polynomials in reduced Gröbner basis (Hilbert's Nullstellensatz)
• Find irreducible components of algebraic varieties through primary decomposition of ideals corresponding to components
• Compute intersection of algebraic varieties by forming ideal generated by defining polynomials and computing Gröbner basis (Elimination Theorem)

Dimension and degree of varieties

• Dimension of algebraic variety defined as dimension of tangent space at smooth point
  • Compute using Gröbner bases: dimension = number of variables - number of polynomials in reduced Gröbner basis
• Degree of algebraic variety counts number of points in intersection with generic linear subspace of complementary dimension
  • Compute using Gröbner bases: degree = number of standard monomials of highest degree w.r.t. lexicographic order
• Examples:
  • Dimension of $V(xy-1)$ in $\mathbb{A}^2$ is 1 since reduced Gröbner basis is $\{xy-1\}$ with 2 variables and 1 polynomial
  • Degree of $V(x^2+y^2-1)$ in $\mathbb{A}^2$ is 2 since highest degree standard monomials are $x$ and $y$

Irreducible components of varieties

• Irreducible components are maximal irreducible subvarieties
  • Algebraic variety is union of its irreducible components
  • Irreducible components uniquely determined
• Find irreducible components using Gröbner bases:
  1. Primary decomposition: decompose ideal of variety into primary ideals, each corresponding to irreducible component
  2. Radical ideals: radical of ideal is intersection of its minimal prime ideals, corresponding to irreducible components
• Example: $V(xy)$ in $\mathbb{A}^2$ has irreducible components $V(x)$ and $V(y)$ since $\langle xy \rangle = \langle x \rangle \cap \langle y \rangle$

Intersections of algebraic varieties

• Intersection of algebraic varieties is set of points belonging to all given varieties
  • Compute intersection using Gröbner bases:
    1. Form ideal generated by polynomials defining varieties
    2. Compute Gröbner basis of ideal
    3. Intersection is variety defined by Gröbner basis
• Elimination theory: eliminate variables using Gröbner bases
  1. Compute Gröbner basis w.r.t. elimination order
  2. Polynomials in Gröbner basis not involving eliminated variables define projection of variety onto remaining variables
• Examples:
  • Intersection of $V(x-y)$ and $V(x+y-1)$ in $\mathbb{A}^2$ is $V(x-\frac{1}{2}, y-\frac{1}{2})$
  • Projecting $V(x^2+y^2-1, z-x)$ onto $xy$-plane eliminates $z$, yielding $V(x^2+y^2-1)$