9.3 Intersection theory for plane curves

Intersection theory for plane curves is a powerful tool in algebraic geometry. It helps us understand how curves intersect, counting their meeting points and measuring how they touch. This theory connects deeply to curve degrees and shapes.

Bézout's theorem is the star of intersection theory. It tells us that two curves' total intersections equal their degree product. This simple idea unlocks complex geometric problems and helps classify curves by their genus and singularities.

Intersection multiplicity of curves

Definition and properties

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• The intersection multiplicity of two plane curves C and D at a point P is a non-negative integer that measures the degree of tangency between the curves at P
• If the curves C and D do not pass through P, the intersection multiplicity at P is 0
• If P is a simple point (transversal intersection) of both curves, the intersection multiplicity is 1
• Higher intersection multiplicities occur when the curves are tangent or have higher-order contact at P (osculating curves, curves with a common cusp)

Computation methods

• The intersection multiplicity can be computed using the dimensions of certain local ring quotients (local intersection number)
• The intersection multiplicity can also be determined by the order of vanishing of resultants (resultant-based approach)
• The sum of intersection multiplicities at all points of intersection of C and D is equal to the product of their degrees, assuming C and D have no common components (Bézout's theorem)

Bézout's theorem for intersections

Statement and proof

• Bézout's theorem states that if C and D are two plane curves of degrees m and n, respectively, with no common component, then the total number of intersections of C and D, counted with multiplicity, is equal to mn
• The proof of Bézout's theorem can be done using resultants, by considering the intersection of C and D as a divisor on the complex projective plane (divisor-based proof)
• Bézout's theorem is a fundamental result in intersection theory and has numerous applications in algebraic geometry and related fields (enumerative geometry, solving systems of polynomial equations)

Applications and examples

• Bézout's theorem can be used to determine the maximum number of intersections between two curves of given degrees (degree bounds)
• The theorem helps analyze the behavior of intersections under certain transformations, such as projective transformations or birational maps (invariance properties)
• Examples of Bézout's theorem include finding the number of intersections between a line and a conic (2 intersections), or between two conics (4 intersections, counting multiplicity)

Intersection theory in geometry

Solving geometric problems

• Intersection theory can be used to determine the number and nature of intersections between plane curves, which is useful in solving various geometric problems
• By applying Bézout's theorem, one can find the total number of intersections (real and complex) between two curves, given their degrees
• Intersection multiplicities can help identify tangency points, multiple points, or other special intersection points between curves (singularities, inflection points)

Studying families of curves

• Intersection theory can be used to study the geometry of families of curves, such as pencils of curves or linear systems (base points, multiplicities)
• Applications of intersection theory include finding common tangents to curves, studying singular points of curves, and analyzing the behavior of curves under certain transformations (projective, birational)
• Intersection theory helps classify special families of curves, such as rational curves, elliptic curves, or curves with prescribed singularities (cuspidal curves, nodal curves)

Intersection theory and curve genus

Genus and its properties

• The genus of a smooth, irreducible, projective plane curve C is a non-negative integer that measures the complexity of the curve's topology (number of holes or handles)
• The genus can be computed using the degree-genus formula: g = (d-1)(d-2)/2, where d is the degree of the curve C
• The degree-genus formula can be derived using intersection theory, specifically by considering the intersection of C with a line and applying the adjunction formula (canonical divisor, Riemann-Roch theorem)

Relation to singularities and classification

• The genus is related to the number and types of singularities (multiple points, cusps, etc.) that a curve possesses; intersection theory helps analyze these singularities (blow-ups, resolution of singularities)
• The genus is an important invariant in the classification of algebraic curves and surfaces, and intersection theory plays a crucial role in studying the geometry and arithmetic of curves (moduli spaces, Jacobians)
• Examples of curves with different genera include rational curves (genus 0), elliptic curves (genus 1), and hyperelliptic curves (genus ≥ 2)