الفهرس 
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Abstract
Currently, algebraic curves and geometry are being applied to areas such as cryptography,
complexity and coding theory, robotics, biological networks, and coupled dynamical systems.
One of the classical topics in the study of algebraic geometry is the study of Weierstrass
points on algebraic curves. The study of Weierstrass points is a fascinating topic, very
rich in geometrical applications and still a ourishing area of research. Weierstrass points
contributed to the understanding of some aspects of the geometry of compact Riemann
surfaces and algebraic curves during the last century from the statement of K. Weierstrass
of the Gap Theorem.
A Weierstrass point on a smooth projective plane curve C of genus g 2 is a point for
which there exists a rational function on C with only a pole at this point of order at most
g and no poles everywhere else. The nite set of Weierstrass points forms an important
invariant of a curve which is of particular use for the study of automorphisms. For example,
using Weierstrass points it can be shown that the automorphism group of a curve is nite.
The Weierstrass points of higher order play also an important role in the theory of alge-
braic curves. For example, one can use them to construct projective embeddings for moduli
spaces of curves. On a curve of genus g 2, Mumford has suggested that the Weierstrass
points of order q (which are called q-Weierstrass points) are analogous to q-torsion points
on an elliptic curve (see §A.III of [33]). The Weierstrass points of higher order are also
important in the arithmetic of algebraic curves and their Jacobians, see for example [6, 39].
from the 1980s onward, the literature on Weierstrass points and related questions started
to grow very fast. In the last 25 years the number of papers which have appeared on this
subject are already more than three times what has been published in the previous thirty. At
the beginning, several researchers developed the theory of the Weierstrass points for smooth
curves, and for their canonical divisors. During the last decades, Lax and Widland, in series
of papers (see [27, 28, 29, 30, 46, 47]) founded and developed the theory for Gorenstein curves,
where the invertible dualizing sheaf replaces the canonical sheaf. Through this context, the
singular points of a Gorenstein curve have to be considered as Weierstrass points.
For more details about the history and the development of the theory of Weierstrass
points and related problems on projective plane curves, we refer, for example [11, 20].
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