![]() In a brilliant lecture on Jthis shy, mentally unstable young man toppled the Euclidean world order once and for all. But a sickly, brilliant mathematician-the second of six children born to a poor Lutheran pastor-Bernhard Riemann (1826-1866) blew the world apart when he proved mathematically that more than three dimensions were not only possible but also highly likely. Until the middle of the last century there was not much talk of a possible Fourth Dimension. Medieval art even accommodated this orthodoxy-paintings were flattened and two dimensional so the viewer could (sort of) see the world as God sees it. God would surely live there, thus He could watch everything that was happening in our 3-D world. Aristotle and Ptolemy added their weight to Euclid by "proving" that any more than three dimensions was "impossible." 1īut of course for those who believed in God, there must be a fourth dimension. Beyond that it was, for centuries, thought impossible for more dimensions to exist. A line had one dimension, a square had two, and a cube, three. The Greek philosopher Euclid (330-275 B.C.) put this down in a mathematical format we now call plane geometry-which ruled the world for the next 2000 years-almost like a religion! In grade school we all learned that the angles of a triangle must add up to 180 degrees, and that a straight line is the shortest distance between two points. We do not have to be told that the world we live in has three dimensions: length, width, and height. Hartshorne, "Algebraic geometry", Springer (1977) MR04631.Born into this world with two eyes, two ears, two arms, two legs and a wonderful data-crunching computer system in both halves of our brains, we humans develop a perception of space as small infants. Zariski, "The theorem of Bertini on the variable singular points of a linear system of varieties" Trans. Nakai, "Note on the intersection of an algebraic variety with the generic hyperplane" Mem. Akizuki, "Theorems of Bertini on linear systems" J. Bertini, "Introduction to the projective geometry of hyperspaces", Messina (1923) (In Italian) The Bertini theorems apply to linear systems of hyperplane sections, without restrictions on the characteristic of the field. If the characteristic of $k$ is finite, the corresponding theorem is true if the extension $k(V)/k(W)$ is separable. If $\dim W = 1$, Bertini's theorem is replaced by the following theorem: Almost all fibres of the mapping $\phi-L : V \to W$ are irreducible and reduced if the function field $k(W)$ is algebraically closed in the field $k(V)$ under the imbedding $\phi_L^* : k(W) \to k(V)$. all except a closed subset in the parameter space $P(L)$ not equal to $P(L)$) are irreducible reduced algebraic varieties.Ģ) Almost all divisors of $L$ have no singular points outside the basis points of the linear system $L$ and the singular points of the variety $V$>.īoth Bertini theorems are invalid if the characteristic of the field is non-zero.Ĭonditions under which Bertini's theorems are valid for the case of a finite characteristic of the field have been studied. The following two theorems are known as the first and the second Bertini theorem, respectively.ġ) If $\dim W > 1$, then almost all the divisors of the linear system $L$ (i.e. Let $V$ be an algebraic variety over an algebraically closed field $k$ of characteristic 0, let $L$ be a linear system without fixed components on $V$ and let $W$ be the image of the variety $V$ under the mapping given by $L$. ![]() ![]() Two theorems concerning the properties of linear systems on algebraic varieties, due to E. ![]()
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