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A linear system of divisors algebraicizes the classic geometric notion of a family of curves, as in the Apollonian circles.. In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family.
As of 2024, Fulton had supervised the doctoral work of 24 students at Brown, Chicago, and Michigan. Fulton is known as the author or coauthor of a number of popular texts, including Algebraic Curves and Representation Theory .
An algebraic curve in the Euclidean plane is the set of the points whose coordinates are the solutions of a bivariate polynomial equation p(x, y) = 0.This equation is often called the implicit equation of the curve, in contrast to the curves that are the graph of a function defining explicitly y as a function of x.
The study of moduli spaces of curves, maps and other geometric objects, sometimes via the theory of quantum cohomology. The study of quantum cohomology, Gromov–Witten invariants and mirror symmetry gave a significant progress in Clemens conjecture. Enumerative geometry is very closely tied to intersection theory. [1]
The properties of algebraic curves, such as Bézout's theorem, give rise to criteria for showing curves actually are transcendental. For example, an algebraic curve C either meets a given line L in a finite number of points, or possibly contains all of L. Thus a curve intersecting any line in an infinite number of points, while not containing ...
Following this purely algebraic problem I would like to raise a question that, it seems to me, can be attacked by the same method of continuous coefficient changing, and whose answer is of similar importance to the topology of the families of curves defined by differential equations – that is the question of the upper bound and position of ...
Trigonal curves include the Picard curves, of genus three and given by an equation y 3 = Q(x) where Q is of degree 4. The gonality conjecture, of M. Green and R. Lazarsfeld, predicts that the gonality of the algebraic curve C can be calculated by homological algebra means, from a minimal resolution of an invertible sheaf of high degree.
In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. [1] The theory for varieties is older, with roots in Bézout's theorem on curves and elimination theory. On the other hand, the topological theory more quickly reached a ...