Ordinary multiple point

Definition

An ordinary multiple point is a type of singular point on a curve where the curve intersects itself with a certain multiplicity. At this point, the derivatives of the curve up to a certain order vanish, indicating that multiple branches of the curve meet at a single point. Understanding ordinary multiple points is crucial when analyzing the behavior of curves and their singularities, especially in determining how these curves can be deformed and what their local structure looks like.

5 Must Know Facts For Your Next Test

1. An ordinary multiple point is specifically defined by having multiplicity greater than one, meaning it represents a situation where two or more branches of the curve converge at the same location.
2. In terms of derivatives, an ordinary multiple point will have all derivatives up to some order vanish at that point, indicating the presence of self-intersection.
3. The local behavior of curves near an ordinary multiple point can often be analyzed using parametrizations to reveal how the branches intersect.
4. Ordinary multiple points can affect intersection theory and are important in classifying curves based on their singularities.
5. In algebraic geometry, identifying ordinary multiple points helps in understanding the smoothness of the underlying space and its implications for further geometric properties.

Review Questions

• How do ordinary multiple points differ from other types of singular points on a curve?
  • Ordinary multiple points are characterized by having a specific multiplicity greater than one, where multiple branches of the curve intersect at a single point. This contrasts with other singular points, which may have different properties, such as cusps or nodes, where the curve may not behave as regularly. Recognizing these distinctions helps in classifying singularities and analyzing their impact on the overall geometry of curves.
• Discuss how multiplicity influences the classification of singularities in algebraic curves, particularly in relation to ordinary multiple points.
  • Multiplicity plays a critical role in classifying singularities because it indicates how many times branches of a curve intersect at a singular point. Ordinary multiple points have multiplicity greater than one, which means they can significantly alter how curves are analyzed and understood. By examining the multiplicity, mathematicians can determine not just if a singularity is ordinary or otherwise but also how it influences the deformation and structural properties of the curve.
• Evaluate the implications of ordinary multiple points for intersection theory and their role in understanding the geometry of algebraic varieties.
  • Ordinary multiple points have significant implications for intersection theory since they represent scenarios where curves intersect in a more complex manner than simple crossing. Their presence alters how we calculate intersections and can lead to deeper insights into geometric structures within algebraic varieties. By studying these points, mathematicians can uncover relationships between different curves and their singularities, enhancing our understanding of algebraic geometry as a whole.

Related terms

Singular Point: A point on a curve where the curve fails to be smooth; for example, where the tangent is not well-defined.

Multiplicity: The number of times a particular root appears in the factorization of a polynomial, reflecting how many times a curve intersects itself at a point.

Tangent Cone: The geometric object that represents all possible tangent lines at a point on a curve, which helps in understanding its local structure around that point.