Question

The object of this problem is to find a property of the local ring \(\mathscr{O}_{P}(F)\) that determines whether or not \(P\) is an ordinary multiple point on \(F\). Let \(F\) be an irreducible plane curve, \(P=(0,0), m=m_{P}(F)>1\). Let \(\mathfrak{m}=\mathfrak{m}_{P}(F)\). For \(G \in k[X, Y]\), denote its residue in \(\Gamma(F)\) by \(g ;\) and for \(g \in \mathfrak{m}\), denote its residue in \(\mathfrak{m} / \mathrm{m}^{2}\) by \(\bar{g}\). (a) Show that the map from \\{forms of degree 1 in \(k[X, Y]\\}\) to \(\mathfrak{m} / \mathrm{m}^{2}\) taking \(a X+\) \(b Y\) to \(\overline{a x+b y}\) is an isomorphism of vector spaces (see Problem 3.13). (b) Suppose \(P\) is an ordinary multiple point, with tangents \(L_{1}, \ldots, L_{m}\). Show that \(I\left(P, F \cap L_{i}\right)>m\) and \(\bar{l}_{i} \neq \lambda \overline{l_{j}}\) for all \(i \neq j\), all \(\lambda \in k\). (c) Suppose there are \(G_{1}, \ldots, G_{m} \in k[X, Y]\) such that \(I\left(P, F \cap G_{i}\right)>m\) and \(\bar{g}_{i} \neq \lambda \bar{g}_{j}\) for all \(i \neq j\), and all \(\lambda \in k\). Show that \(P\) is an ordinary multiple point on \(F .\) (Hint:: Write \(G_{i}=L_{i}+\) higher terms. \(\bar{l}_{i}=\bar{g}_{i} \neq 0\), and \(L_{i}\) is the tangent to \(G_{i}\), so \(L_{i}\) is tangent to \(F\) by Property (5) of intersection numbers. Thus \(F\) has \(m\) tangents at \(P .\) ) (d) Show that \(P\) is an ordinary multiple point on \(F\) if and only if there are \(g_{1}, \ldots, g_{m} \in \mathfrak{m}\) such that \(\bar{g}_{i} \neq \lambda \bar{g}_{j}\) for all \(i \neq j, \lambda \in k\), and \(\operatorname{dim} \mathscr{O}_{P}(F) /\left(g_{i}\right)>m\)

Short answer

Question: Provide a necessary and sufficient condition for P to be an ordinary multiple point on F in terms of the local ring O_P(F). Answer: P is an ordinary multiple point on F if and only if there exist g_1, ..., g_m in the ideal 𝔪 such that their images in 𝔪/𝔪^2, denoted by 𝑏𝑎𝑟(𝑔𝑖), are distinct and non-zero, i.e. 𝑏𝑎𝑟(𝑔𝑖) ≠ 𝜆𝑏𝑎𝑟(𝑔𝑗) for all i ≠ j and λ ∈ k, and the dimension of the quotient ring 𝒪_P(F)/(g_i) is greater than m, i.e. dim 𝒪_P(F)/(g_i) > m.

Step-by-step solution

(a) Proving the Isomorphism

First, we need to show that the given map is an isomorphism between the vector spaces of forms of degree 1 in \(k[X, Y]\) and \(\mathfrak{m}/\mathfrak{m}^2\). Take any \(aX + bY\) in \(k[X, Y]\) and its image \(\overline{ax + by}\) as defined. To prove the isomorphism, we need to show that the map is injective (one-to-one) and surjective (onto). Injectivity: Suppose \(\phi(aX+bY) = \phi(cX+dY)\). This implies that \(\overline{ax + by} = \overline{cx + dy}\). Hence, \((ax + by) - (cx + dy) \in \mathfrak{m}^2\), which means \(aX + bY - cX - dY \in (X, Y)^2\). Thus, \((a-c)X + (b-d)Y \in (X, Y)^2\). But this is possible only if \((a-c) = 0\) and \((b-d) = 0\), so \(a=c\) and \(b=d\). Therefore, our map is injective. Surjectivity: Given any element \(\bar{g} \in \mathfrak{m}/\mathfrak{m}^2\), we know it corresponds to some \(g=ax+by \in \mathfrak{m}\). We can then show that our map sends \(aX+bY\) to \(\bar{g}\): \(\phi(aX+bY) = \overline{ax + by} = \bar{g}\). Thus, our map is surjective. Since the map is injective and surjective, it is an isomorphism of vector spaces.

(b) Properties for Ordinary Multiple Points

Suppose \(P\) is an ordinary multiple point on \(F\) with tangents \(L_1, \ldots, L_m\). We need to show that \(I\left(P, F \cap L_{i}\right) > m\) and \(\bar{l}_i \neq \lambda \bar{l}_j\) for all \(i \neq j\) and all \(\lambda \in k\). Consider the intersection multiplicities for \(F \cap L_{i}\). Since each \(L_i\) is tangent to \(F\) at \(P\), we know \(I(P, F \cap L_i) = m_i\) where \(m_i\) is the intersection number between \(F\) and \(L_i\) at \(P\). But for ordinary multiple point \(P\), \(F\) has distinct tangents. Therefore, the intersection number must be greater than the multiplicity \(m\) of \(P\). Hence we have \(I\left(P, F \cap L_{i}\right) > m\). Now we need to show that \(\bar{l}_i \neq \lambda \bar{l}_j\) for all \(i \neq j\) and all \(\lambda \in k\). Since \(L_i\) is a form of degree 1 in \(k[X,Y]\), by part (a), we know \(\bar{l_i}\) correspond to different non-zero elements in \(\mathfrak{m}/\mathfrak{m}^2\). So, \(\bar{l}_i \neq \lambda \bar{l}_j\) for all \(i \neq j\) and all \(\lambda \in k\).

(c) Conditions Implying P is an Ordinary Multiple Point

Suppose there exist \(G_1,\ldots,G_m \in k[X,Y]\) satisfying the given conditions. Define \(L_i\) as the tangent to \(G_i\) at \(P\). Since \(\bar{l}_i=\bar{g}_i\neq 0\), each \(G_i\) has a well-defined tangent line at \(P\). We know \(I(P,F\cap G_i)>m\) for each \(i\). By Property (5) of intersection numbers: if two curves \(F_1\) and \(F_2\) have \(I(P, F_1 \cap F_2) > \operatorname{mult}_P(F_1) \cdot \operatorname{mult}_P(F_2)\), then \(F_1\) and \(F_2\) have a common tangent at \(P\). Let \(F_1=F\) and \(F_2=G_i\), then we have \(I(P,F \cap G_i) > m \cdot 1\). This implies that \(F\) and \(G_i\) have a common tangent \(L_i\) at \(P\). Thus, we have a tangent \(L_i\) at \(P\) for each \(G_i\). Our elements \(\bar{g}_i \neq \lambda \bar{g}_j\) for all \(i \neq j\) and all \(\lambda \in k\). This implies that \(F\) has \(m\) distinct tangents at \(P\). Consequently, \(P\) is an ordinary multiple point on \(F\).

(d) Necessary and Sufficient Condition for Ordinary Multiple Points

We need to show that \(P\) is an ordinary multiple point on \(F\) if and only if there are \(g_1, \ldots, g_m \in \mathfrak{m}\) such that \(\bar{g}_i \neq \lambda \bar{g}_j\) for all \(i \neq j, \lambda \in k\), and \(\operatorname{dim} \mathscr{O}_P(F) /(g_i) > m\). Part (b) showed that if \(P\) is an ordinary multiple point on \(F\), then there exist \(g_i \in \mathfrak{m}\) satisfying the required conditions. Part (c) showed that if these conditions are satisfied, then \(P\) must be an ordinary multiple point on \(F\). Together, these results prove that the necessary and sufficient condition for \(P\) to be an ordinary multiple point is that there must exist \(g_1,\ldots,g_m \in \mathfrak{m}\) such that \(\bar{g}_i \neq \lambda \bar{g}_j\) for all \(i \neq j, \lambda \in k\), and \(\operatorname{dim} \mathscr{O}_P(F) /(g_i) > m\).

Key concepts

Local Ring

In algebraic geometry, a local ring is a crucial structure at a specific point on a curve or surface, capturing the local behavior of functions around that point.
A local ring is particularly essential when examining an irreducible plane curve like\(F\) at a point\(P\).
A local ring \(\mathscr{O}_{P}(F)\) is tailored to know everything about the curve \(F\) near \(P\), ignoring the curve's behavior elsewhere. It consists of germs of functions at that particular point, meaning the functions are defined around the point \(P\).

• The maximal ideal \(\mathfrak{m}_{P}(F)\) is in \(\mathscr{O}_{P}(F)\) and consists of all elements that vanish at \(P\).
• The residue field \(k\) often represents the functions that don't vanish at all.

This local ring offers special insight on ordinary multiple points, as its properties can show whether \(P\) is indeed such a point on \(F\). By examining how the vector space \(\mathfrak{m}_{P}(F) / \mathfrak{m}_{P}(F)^2\) behaves, we gain insights into the tangent lines at \(P\).
In doing so, we understand how \(P\) might link to being an ordinary multiple point on the curve.

Intersection Numbers

Intersection numbers are a vital tool in understanding how curves intersect at points. At their core, they quantify the number of times two curves meet at a given location.
When studying a point \(P\) on an irreducible plane curve \(F\), intersection numbers help assess the multiplicity of intersections. This is paramount when determining if \(P\) is an ordinary multiple point. An intersection number greater than the point's multiplicity suggests more overlap than a simple crossing.

• An intersection number \(I(P, F \cap L_i) > m\) indicates that the line \(L_i\) is indeed tangent to \(F\) at \(P\), implying \(P\) is an ordinary multiple point if this holds for multiple lines.
• Property (5) of intersection numbers asserts that if two curves have an intersection number exceeding the product of their local multiplicities, they share a tangent.

Thus, intersection numbers offer crucial evidence needed to establish the nature of the points on a curve and are a guiding factor in classifying intersections as ordinary or more complex.

Tangent Lines

Tangent lines are straight lines that merely "touch" a curve at a given point without crossing it at that point. They provide insightful geometric information about the curve and are integral when studying ordinary multiple points.
At a suspected ordinary multiple point \(P\), an irreducible plane curve \(F\) is expected to have several tangent lines \(L_1, L_2, \ldots, L_m\).

• If \(I(P, F \cap L_i) > m\), it indicates \(L_i\) is tangent to \(F\) at \(P\), meaning \(P\) could be an ordinary multiple point.
• Distinctness of the tangents \(L_i\) at \(P\) is essential; no line can be a scalar multiple of another within the residue class \(\mathfrak{m} / \mathfrak{m}^2\).

Tangent lines at \(P\) ensure that each line \(L_i\) touches the curve only where they appear tangentially; they do not cross \(F\) at \(P\) but may extend to interact with it elsewhere.

Irreducible Plane Curve

An irreducible plane curve is a geometric object defined by a polynomial equation where the curve cannot be further decomposed into simpler components over the field of coefficients.
Such a curve represents the entirety of its path in the plane and doesn't break into smaller, non-intersecting curves.

• It remains intact even under complex transformations, making it a single, unified curve entity.
• Ordinary multiple points reside on these curves, where the curve locally behaves as if it were multiple distinct lines overlapping.

Exploring properties like the local ring at a point \(P\) on an irreducible plane curve \(F\) further reveals the complexity and depth of the curve's construction. Distinguishing ordinary multiple points becomes a study of the interplay between algebraic structures and geometric intuition based on the curve's behavior. These curves are fundamental objects in algebraic geometry, underlying fundamental properties like tangent lines and intersection numbers.