TL;DR
This paper investigates the problem of determining the number of surfaces of a certain degree intersecting a given surface in P^3 with specified multiplicities at general points, proposing conjectures and proving them in cases of small multiplicities.
Contribution
It formulates two conjectures related to surface intersections in P^3 and proves their validity for small multiplicities, advancing understanding of intersection theory.
Findings
Conjectures are equivalent in the context of surface intersections.
Proved the conjectures for cases with small multiplicities.
Provides a framework for future research on intersection counts.
Abstract
Given a surface S in P^3 and a collection of general points on it, how many surfaces of a given degree intersect S in a curve with prescribed multiplicities at the points? We formulate two natural conjectures which would answer this question, and we show they are equivalent. We then prove these conjectures in case all multiplicities are small.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Polynomial and algebraic computation · Vietnamese History and Culture Studies