What is algebraic geometry?
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Algebraic geometry has a long history that can be said to go back to the Euclidean geometry in ancient Greece. A milestone was Descartes' introduction of coordinates in geometry. This meant that one could go back and forth between an algebraic treatment of equations for a geometric object and a direct geometric study of the object. Today, algebraic geometry is an area with geometry with connections to other areas such as commutative algebra, complex analysis, topology and number theory in mathematics, cryptography in informatics and string theory in physics.
The objects we study in algebraic geometry are algebraic varieties, which we can say er geometric objects that can be defined by solution sets of polynomials. For example, the two-dimensional sphere can be defined in the three-dimensional Euclidean space R3 as the set of all points (x,y,z) that satisfy x2+y2+z2-1=0.
If we look further at all the points the satisfy the two equations x2+y2+z2-1=0 and x+y+z=0, then we get the circle at an angle marked in red on the figure.
On the other hand, modern algebraic geometry is not so concerned about the equations that describe the objects, but concentrates on the abstract properties of the geometric objects by assigning them algebraic structures. The translation to algebra means that algebraic geometry is more suitable for studying geometric problems of higher complexity than other nearby fields. This applies not least to the study of singular points on geometric objects, which are, broadly speaking, the points where the object is not "smooth". A classic example is the origin of the curve defined by the equations x3-y2=0.
