Question

Let \(P=[0: 1: 0], F\) a curve of degree \(n, F=\sum F_{i}(X, Z) Y^{n-i}, F_{i}\) a form of degree \(i\). Show that \(m_{P}(F)\) is the smallest \(m\) such that \(F_{m} \neq 0\), and the factors of \(F_{m}\) determine the tangents to \(F\) at \(P\).

Short answer

Answer: The multiplicity of the intersection of a curve F at a point P, denoted by m_P(F), is the smallest m such that F_m ≠ 0. It represents the order of vanishing of the function F at P. The factors of F_m determine the tangents to the curve F at the point P.

Step-by-step solution

Determine the multiplicity \(m_P(F)\) of the intersection at \(P\)

Recall that the multiplicity of the intersection of a curve \(F\) at a point \(P\), \(m_P(F)\), is the order of vanishing of the function \(F\) at \(P\). That is, how many times the function needs to be differentiated (with respect to either \(X\) or \(Z\)) before it becomes non-zero at the point \(P\). We can express this mathematically as: $$ m_P(F) = \min_{i}\{i : F_i(0, 0) \neq 0\} $$

Find the smallest \(m\) for which \(F_m \neq 0\)

To find the smallest \(m\) such that \(F_m \neq 0\), we substitute the coordinates of the point \(P=[0:1:0]\) into the curve equation: $$ F(P) = \sum_{i=0}^{n} F_i(0,0) \cdot Y^{n-i} $$ Since \(Y=1\) at point \(P\), this equation simplifies to: $$ F(P) = \sum_{i=0}^{n} F_i(0,0) $$ Now, we can find the smallest \(m\) for which \(F_m \neq 0\) by enumerating the terms in the sum: $$ F_0(0,0) = 0, F_1(0,0) = 0, \dots, F_m(0,0) \neq 0 $$ Thus, \(m_P(F)\) is the smallest \(m\) such that \(F_m \neq 0\).

Analyze the factors of \(F_m\) and show they determine the tangents to \(F\) at the point \(P\)

Since \(F_m \neq 0\), we can now factorize \(F_m\) into its irreducible components: $$ F_m = G_{1}G_{2}\dots G_{k} $$ where \(G_{j}\) are irreducible forms. Let \(t\) be a tangent to \(F\) at \(P\). Since \(m_P(F)\) is the multiplicity of the intersection, Bezout's theorem implies that \(F\) and \(t\) intersect at \(P\) with multiplicity at least \(m_P(F)\). Thus, the tangent \(t\) must be a divisor of \(F_m\). Therefore, the tangents to the curve \(F\) at the point \(P\) are determined by the factors of \(F_m\).

Key concepts

Multiplicity

In algebraic geometry, multiplicity is an essential concept that helps us understand how a curve intersects with a point. Multiplicity can be thought of as the number of times a curve touches or intersects a point before moving off in another direction. We calculate this by considering the order of vanishing of the function at the point.

• If a curve intersects a point without changing direction (like just touching the point), it has a higher multiplicity.
• If it simply passes through the point, the multiplicity is lower.

Thus, multiplicity is a measure of how tangentially the curve approaches a point. For a curve represented by a polynomial equation, finding the multiplicity involves determining how many times you would differentiate the function until it no longer equals zero at that point.
In our exercise, we saw that the multiplicity of intersection at point \(P\) (notated as \(m_P(F)\)) is determined by the smallest degree of \(i\) for which \(F_i(0, 0) eq 0\). This tells us how the curve approaches the specific point \(P\).

Intersection Multiplicity

Intersection multiplicity gives us more insight into how two curves behave when they meet at a given point. Just like singular multiplicity, intersection multiplicity counts the 'intensity' of their meeting.

• An intersection with a higher multiplicity means the curves not only meet but also adhere more closely together.
• Conversely, a lower intersection multiplicity signifies a more casual crossing.

We determine the intersection multiplicity at a point by taking the overlap of common factors present in the polynomial expressions defining each curve.
In the context of our exercise, we're concerned with the multiplicity of the intersection of the curve \(F\) at the point \(P\). Specifically, it involves analyzing for the smallest order \(m\) such that the polynomial terms \(F_m\) do not vanish, revealing the richness of the intersection.

Irreducible Components

Irreducible components are integral to understanding the structure within a polynomial curve. An irreducible component is a factor of a polynomial that cannot be broken down or factored further into simpler polynomials over its coefficient field.

• Each component stays as it is under factorization over the given field.
• This helps in identifying the basic building blocks of curves.

In the exercise, finding the smallest \(m\) such that \(F_m eq 0\) lets us express \(F_m\) as a product of these irreducible components. Each component represents a potential tangent to the curve at the intersection point.
Thus, irreducible components inform us of how the curve could branch out and help determine the tangents at the given point \(P\).

Bezout's Theorem

Bezout's Theorem is a fundamental principle in algebraic geometry that gives us a powerful way to predict the intersection behavior of curves. It states that if two algebraic curves of degrees \(m\) and \(n\) intersect, the total number of intersection points (counted with their respective multiplicities) is exactly \(m \cdot n\), as long as we're in a projective plane and curves are not defined over special points.

• This theorem helps determine possible intersection scenarios for curves.
• It emphasizes that for tangents, the intersection multiplicity plays a key role.

In the discussed exercise, Bezout’s theorem underpins the relationship between the curve \(F\) and its tangents at \(P\). Since \(t\), as a tangent, divides \(F_m\), by Bezout’s theorem the intersection at \(P\) must account for the multiplicity \(m_P(F)\). Therefore, the theorem is essential in defining how factors of \(F_m\) determine the tangents, helping understand deeper geometric relationships of curves at intersections.