Question

Let \(V \subset \mathbb{A}^{n}\) be an affine variety, \(P \in V .\) The tangent space \(T_{P}(V)\) is defined to be \(\left\\{\left(v_{1}, \ldots, v_{n}\right) \in \mathbb{A}^{n} \mid\right.\) for all \(\left.G \in I(V), \sum G_{X_{i}}(P) v_{i}=0\right\\} .\) If \(V=V(F)\) is a hypersurface, \(F\) irreducible, show that \(T_{P}(V)=\left\\{\left(v_{1}, \ldots, v_{n}\right) \mid \sum F_{X_{i}}(P) v_{i}=0\right\\} .\) How does the dimension of \(T_{P}(V)\) relate to the multiplicity of \(F\) at \(P\) ?

Short answer

Question: Prove that the tangent space of a hypersurface with an irreducible function at a given point is equal to the set of vectors for which the partial derivatives of the function at that point, when multiplied by the components of the vectors, sum to zero. Also, analyze how the dimension of the tangent space relates to the multiplicity of the irreducible function at the given point. Answer: We proved that for a hypersurface defined by an irreducible function \(F\), the tangent space \(T_P(V)\) at a given point \(P\) is equal to the set of vectors for which the partial derivatives of the function, when multiplied by the components of the vectors, sum to zero. The dimension of the tangent space \(T_P(V)\) is related to the multiplicity \(m\) of the irreducible function \(F\) at the point \(P\) by: $$\dim T_P(V) \le n - m$$ where \(n\) is the dimension of the ambient space, and the equality holds when the derivatives of \(F\) of order \(m-1\) are linearly independent at point \(P\). This means that the dimension of the tangent space decreases as the multiplicity of the function increases, as a higher multiplicity corresponds to a curve bending more around that point.

Step-by-step solution

Definitions

Affine Variety: An affine variety \(V \subset \mathbb{A}^n\) is a solution set of a system of polynomial equations. Tangent Space: The tangent space \(T_P(V)\) at a point \(P \in V\) can be thought of as the space representing all possible tangent directions to the variety at that point. More precisely, in this exercise, the tangent space is defined as the set \(\{(v_1, \dots, v_n) \mid \sum G_{X_i}(P)v_i = 0\}\) where \(G \in I(V)\) is a polynomial belonging to the ideal of the variety. Hypersurface: A hypersurface is a variety defined by the vanishing set of a single irreducible polynomial, i.e., \(V(F)\) where \(F\) is an irreducible polynomial. Multiplicity: The multiplicity of a polynomial \(F\) at a point \(P\) is the number of times the polynomial "crosses" that point, counting tangency. Algebraically, this is given by the order of vanishing of the polynomial at that point.

Prove the Equality for Hypersurface

Let \(V = V(F)\) be a hypersurface, and \(F\) be an irreducible polynomial. We want to show that for any point \(P \in V\), the tangent space \(T_P(V)\) is given by the set \(\{(v_1, \dots, v_n) \mid \sum F_{X_i}(P)v_i = 0\}\). Since \(V\) is a hypersurface, it is defined by a single polynomial, and the ideal of the variety is \(I(V) = (F)\). Now, consider any \(G \in I(V)\). Since \(G\) belongs to the ideal generated by \(F\), we can write \(G = aF\) for some polynomial \(a\). To show that the tangent spaces are equal, we need to prove two inclusions: Claim: \(\{(v_1, \dots, v_n) \mid \sum G_{X_i}(P)v_i = 0\} \subset \{(v_1, \dots, v_n) \mid \sum F_{X_i}(P)v_i = 0\}\) Take an arbitrary vector \((v_1, \dots, v_n)\) from the left-hand set, i.e., \(\sum G_{X_i}(P)v_i = 0\). Since \(G = aF\), we have \(\sum (aF)_{X_i}(P)v_i = 0\). Differentiating the product \(aF\) with respect to \(X_i\), we obtain: $$(aF)_{X_i} = a_{X_i}F + aF_{X_i}$$ Evaluating this expression at point \(P\), we get: $$\sum (a_{X_i}(P)F(P) + a(P)F_{X_i}(P))v_i = 0$$ However, since \(P \in V\), we have \(F(P) = 0\) and thus: $$\sum a(P)F_{X_i}(P)v_i = 0$$ Thus, the vector \((v_1, \dots, v_n)\) belongs to the right-hand set as well, proving the first inclusion. Claim: \(\{(v_1, \dots, v_n) \mid \sum F_{X_i}(P)v_i = 0\} \subset \{(v_1, \dots, v_n) \mid \sum G_{X_i}(P)v_i = 0\}\) Since \(G \in I(V)\), and \(I(V) = (F)\), the reverse inclusion can be proven similarly to the first inclusion, and thus both sets are equal.

Analyzing the Relation Between Tangent Space Dimension and Multiplicity

Let \(m\) be the multiplicity of \(F\) at point \(P\). If \(F(P) = 0\), its partial derivatives of order less than \(m\) also vanish at \(P\), meaning that the constraint \(\sum F_{X_i}(P)v_i = 0\) is stronger for higher multiplicities. Intuitively, if a polynomial crosses a point with higher multiplicity, its tangent space has a smaller dimension as the curve "bends" more around that point. More formally, it can be shown that for a point \(P\) on a hypersurface \(V\), the dimension of the tangent space \(T_P(V)\) is related to the multiplicity of \(F\) at \(P\) by: $$\dim T_P(V) \le n - m$$ where \(n\) is the dimension of the ambient space, and \(m\) is the multiplicity of \(F\) at \(P\). The equality holds when the derivatives of \(F\) of order \(m-1\) are linearly independent at point \(P\).

Key concepts

Algebraic Geometry

Algebraic geometry is a branch of mathematics that combines abstract algebra, especially commutative algebra, with geometry. It allows us to study the solutions of systems of polynomial equations by viewing them as geometric spaces called varieties. An affine variety, which is a fundamental concept in this field, is formed by the set of solutions to a system of algebraic equations over the complex numbers, denoted by \( \bb{C} \). In simpler terms, if you have one or more polynomials, the collection of points where these polynomials simultaneously equal zero constitutes an affine variety.

This relationship between algebra and geometry becomes incredibly powerful as it lets us visualize and work with complex algebraic structures. For instance, when investigating the properties of a variety, we can consider aspects such as dimension, meaning how 'big' the variety is in the geometrical sense, and tangent spaces, which tell us about the behavior of the variety in the immediate vicinity of a given point.

Hypersurface

A hypersurface in algebraic geometry is a specific type of variety defined by a single irreducible polynomial equation. Essentially, it is an object of codimension one in the ambient space, meaning that it is one dimension less than the space it lies in. If the ambient space is \( \bb{A}^n \), then a hypersurface in this space would typically have dimension \(n-1\).

For example, consider a simple three-dimensional space, \( \bb{A}^3 \). A hypersurface in this context could be a curve (1D) or a surface (2D). The irreducibility of the defining polynomial is crucial as it ensures that the hypersurface is not comprised of 'smaller' pieces or components. When dealing with hypersurfaces, the study of their properties, such as their tangent spaces, becomes especially compelling because of the geometric intuition that a single equation provides.

Polynomial Multiplicity

Polynomial multiplicity is a concept that pertains to the roots of polynomials. Multiplicity refers to the number of times a particular root is repeated for a polynomial. For example, if a polynomial \(f(x)\) has a root \(x=p\) with multiplicity \(m\), then \(f(x)\) can be factored as \( (x-p)^m g(x)\), where \(g(p) \eq 0\).

In the setting of algebraic geometry, multiplicity has a geometric interpretation at points on a variety. If a polynomial defining a hypersurface vanishes at a point with multiplicity \(m\), it's akin to saying that the 'curve' associated with the polynomial 'touches' the point repeatedly \(m\) times. High multiplicity at a point generally leads to a lower-dimensional tangent space at that point because the polynomial 'holds on' to the variety more tightly; the hypersurface bends sharply around these points, restricting the directions in which the tangent can move. A compelling aspect of this is its counter-intuitive nature—the 'sharper' or 'tighter' the curve at a point, the 'smaller' the set of tangent directions.