WDS'08 Proceedings of Contributed Papers, Part I, 32–37, 2008. ISBN 978-80-7378-065-4 © MATFYZPRESS

Luigi and his Transformations

D. Trkovsk´a , Faculty of Mathematics and Physics, , .

Abstract. The present paper is dedicated to Luigi Cremona (1830–1903), one of the leading Italian mathematicians of the second half of the 19th century. At first, it gives a brief account of his professional life. In the following text, we describe fundamental results of the theory of birational transformations, later known as Cremona transformations. At the end, we discuss Cremona’s influence on the Czech geometrical school, with main attention to (1848–1894) who spent some time in attending Cremona’s lectures. References to selected papers relevant to this theme are attached.

Introduction Luigi Cremona had a great influence on Italian , he was one of the founders of the new Italian mathematical school. He was mainly interested in projective and algebraic geometry and discovered graphical methods for solving problems in statics as well. Many propositions on synthetic geometry were revised and improved by him. Birational transformations of a projective plane or space named after Cremona play an important role in algebraic geometry.

Professional Life of Luigi Cremona Luigi Cremona was born on December 7, 1830 in , (today Italy). Having finished his study at the grammar school in Pavia, he was made by political events of the revolution of 1848 to interrupt his further studies. As an Italian nationalist he joined the Free Italy battalion and took part in the defence of against the Austrian army. Although Cremona with his troops had to surrender in August 1849, their bravery had been such that the Austrian attackers allowed them to leave the city with an honour. Then, Cremona returned back to Pavia and entered the to study for a degree in civil engineering. During that time he was mostly influenced by his teacher (1824–1897). By that time Camillo Benso di Cavour (1810–1861) had become the head of the government of Piemont and the unification of Italy had begun. However, the region of Lombardy was still controlled by the Austrian army. Cremona, who had been fighting against the Austrian occupation, could not obtain an official teaching post and therefore he served at first as a private teacher in several families in Pavia. At that time Cremona was also engaged in mathematical research; his first mathematical paper was published in March 1855. It had a great effect on his future because it helped him to gain a permission to teach physics on a temporary basis at the grammar school in Pavia where he had himself been educated. In May 1856, his second mathematical paper appeared and this, together with his great reputation as a teacher, helped him to gain the position of an associate professor in December 1856. Consequently, in January 1857, Cremona was appointed as a full professor at the grammar school in Pavia. He stayed there for three years and during that time he wrote a number of mathematical papers on curves using methods of projective geometry. In 1859, Italy in the alliance with France got free from the Austrian rule, Lombardy was liberated and Cremona should no longer be held back for political reasons. On November 28, 1859 he began to teach at the Lyceum of Saint Alexander in and on June 10, 1860 he was appointed by the Royal Decree as an ordinary professor at the and served there until October 1867. During that time he published about 45 mathematical papers including his most important work on transformations of plane curves which won him

32 TRKOVSKA:´ LUIGI CREMONA AND HIS TRANSFORMATIONS the Steiner Prize1 in 1864. Also while at the University of Bologna, Cremona developed the theory of birational transformations, later known as Cremona transformations. In October 1867, on Brioschi’s recommendation, Cremona was appointed by another Royal Decree as a teacher to the Polytechnical Institute of Milan where he received the Professor title in 1872. This period is thought to be the time of Cremona’s gratest productivity. He wrote many articles on such diverse topics as conic sections, plane curves, developable surfaces, third- and fourth-degree surfaces, projective geometry and also on graphical statics. In 1873, Cremona was offered a political post of general secretary of the new Italian govern- ment. Although he was as an Italian patriot very pleased with this offer, he had to refuse it because of his serious engagement in mathematical research. Instead, on October 9, 1873, he moved to where he was appointed by another Royal Decree as a director and professor of graphical statics of the newly established Polytechnical School of Engineering. However, if he had hoped that refusing of the political post would leave him able to continue his mathematical research, he soon found out that his administration and teaching duties made him very busy and he had no time for his scientific activities. In November 1877, Cremona was appointed to the Chair of higher mathematics at the University of Rome. However, political pressures made him feel that he should serve the new Italian State. On March 16, 1879 Cremona was elected as a Senator and at this point his mathematical work ended completely. In the next years he became the Minister of Public Education and ended his political career as the Vice-President of the Senate. Luigi Cremona died of a heart attack on June 10, 1903 in Rome.

Figure 1. Luigi Cremona with his signature.

In mathematics, Luigi Cremona was interested in projective and algebraic geometry, graphi- cal statics, differential and integral calculus and application of algebraic methods to geometry. During his professional life he published more than one hundred mathematical papers many of which have been translated into other languages. Amongst many other honours, Luigi Cremona was a corresponding member of the Royal Society of London since 1879, and a member of the Royal Society of Edinburgh since 1883.

1Jacob Steiner (1796–1863), a Swiss mathematician who was interested in synthetic and projective geometry.

33 TRKOVSKA:´ LUIGI CREMONA AND HIS TRANSFORMATIONS

He was also a member of academies and scientific associations in Bologna, Napoli, G¨ottingen, Lisbon, Venice and Prague where he became the first honorary foreign member of the Union of Czech Mathematicians in 1871. After his death one of the Moon craters was named Cremona.

Cremona Transformations In the second half of the 19th century, mathematicians all over the world paid attention to the study of geometric transformations. They especially drew their attention to higher degree transformations, the main stress was laid upon birational transformations. A birational transformation of a projective plane is a transformation of the form

x′ = φ(x, y), y′ = ψ(x, y), where φ and ψ are rational functions of non-homogeneous coordinates x and y which can also be expressed rationally in terms of coordinates x′ and y′. In homogeneous coordinates x0, x1 and x2 every birational transformation can be expressed in the form

x′i = Fi(x0, x1, x2) for i = 0, 1, 2; inverse transformations are of the form

xi = Gi(x0′ , x1′ , x2′ ) for i = 0, 1, 2, where Fi and Gi are homogeneous polynomials of the corresponding variables. A Cremona transformation is subsequently defined as any birational transformation of a projective space over a field K. Cremona transformations of the given space form a group which is called the Cremona group. The simplest example of Cremona transformations is circular inversion of a plane which is defined as a transformation under which corresponding points X and X′ are collinear with the fixed centre S of inversion and the product of the distances SX and SX′ is equal to possitive constant r2, where r is the radius of the given circle. If we set up a coordinate system at the centre S of the circle, this transformation may be expressed analytically in the form

r2x r2y x = , y = . ′ x2 + y2 ′ x2 + y2 Circular inversion carries a straight line or circle again into a straight line or circle. It was the first non-trivial geometric transformation which was studied in the context of birational transformations. Another examples of Cremona transformations are quadratic birational transformations of a plane. In non-homogeneous coordinates x, y they may be expressed as linear-fractional functions a1x + b1y + c1 a3x + b3y + c3 x′ = , y′ = , a2x + b2y + c2 a4x + b4y + c4 where ai, bi and ci are real coefficients for i = 1, 2, 3, 4. Among these transformations, special attention is paid to the standard quadratic transformation 1 1 x = , y = , ′ x ′ y or, in homogeneous coordinates,

x0′ = x1x2, x1′ = x0x2, x2′ = x0x1.

34 TRKOVSKA:´ LUIGI CREMONA AND HIS TRANSFORMATIONS

Both circular inversion and quadratic birational transformations are transformations of the second degree.2 Cremona was in his work Sulle trasformazioni geometriche delle figure piane [Geometric Transformations of Plane Figures] from 1863 and 1865 engaged in the question on how to construct general geometric transformation of an arbitrary degree, i.e. how to construct a transformation which carries straight lines into curves of an arbitrary degree. In his work Cremona derived two fundamental equations which must be held for the number of points common to all those curves.

Consider two geometric figures, first in a plane P and the second one in a plane P ′ and any one-to-one transformation between them, i.e. any transformation that carries every point of the figure P into just one point of the figure P ′ and conversely. The question of principle is which curves in the second figure correspond under the transformation to straight lines in the

first figure. Let us denote the degree of a curve in the figure P ′ which corresponds to general straight line in the figure P by the letter n. Every straight line is uniquely determined by two non-identical points, that is why the curve of the degree n is uniquely determined by the images of those two points under the transformation. It implies that curves in the figure P ′ constitute a geometric system of the degree n. Any curve of the degree n is generally determined by n(n+3) independent conditions. Hence, 2 the curves in the figure P ′ which correspond to straight lines in the figure P must comply with n(n+3) − 2 = (n 1)(n+4) common conditions. Two straight lines in the figure P have just one 2 − 2 intersection point and that is why their corresponding curves of the degree n must uniquely determine its corresponding point. But two curves of the degree n have generally n2 common points (including their multiplicities) and it follows that the other n2 − 1 points of intersection must be common to all curves from the thought geometric system. Let us denote the number of points with multiplicity r common to all curves from the thought geometric system by the symbol xr. It is known that a point with multiplicity r which is common to two curves represents r2 simple intersection points and therefore we have the equation 2 2 x1 + 4x2 + 9x3 + 16x4 + ... + (n − 1) xn 1 = n − 1. (1) − All points common to all curves from the thought geometric system represent (n 1)(n+4) common − 2 conditions mentioned above. Every point with multiplicity r represents in the determination of algebraic curves r(r+1) simple conditions and therefore we have the equation 2 n(n − 1) (n − 1)(n + 4) x1 + 3x2 + 6x3 + ... + xn 1 = . (2) 2 − 2 Cremona remarked in his work that the same equations (1) and (2) will be obtained if we consider curves of arbitrary degree instead of straight lines in the figure P . We have proved that the curves corresponding to straight lines under any Cremona trans- formation must have common a certain number of points which are called fundamental points of the transformation. Moreover, the numbers of fundamental points of particular multiplicity must hold the equations (1) and (2). These equations have more solutions in general and every solution defines special Cremona transformation. By subtraction the equation (2) from the equation (1) we obtain the equation (n − 1)(n − 2) (n − 1)(n − 2) x2 + 3x3 + ... + xn 1 = , (3) 2 − 2 from which it follows xn 1 = 0 or xn 1 = 1. In case xn 1 = 1 we have x2 = x3 = ... = xn 2 = 0 − − − − and x1 = 2n − 2.

2The degree of a transformation is defined as the degree of the curve which corresponds under this transfor- mation to general straight line.

35 TRKOVSKA:´ LUIGI CREMONA AND HIS TRANSFORMATIONS

Hence, we can see that amongst all transformations associated with certain value of n is involved one transformation which seems to be the simplest. Under this transformation the curves of the degree n corresponding to straight lines have common only one point of multiplicity n−1 and other 2n−2 simple points. Such transformation is called de Jonqui`eres transformation3 and can be generally expressed in the form

P (x)y + Q(x) x = x, y = , ′ ′ R(x)y + S(x) where P (x), Q(x), R(x), S(x) are polynomials of variable x over a field K. To illustrate Cremona’s results let us present two examples. For n = 2 both equations (1) and (2) reduce to x1 = 3. In this case straight lines transform into curves of the second degree which have common three simple points. This transformation is called conical transformation. For n = 3 equations (1) and (2) imply x1 = 4, x2 = 1. In this case straight lines transform into curves of the third degree which have common one double point and four simple points. One can prove that by composition of an arbitrary sequence of collineations (i.e. linear transformations) and standard quadratic transformations we obtain Cremona transformation as well. General quadratic transformations play moreover a fundamental role in the theory of Cremona transformations. By Noether’s Factorization Theorem:4 If K is an algebraically closed field, any Cremona transformation of the projective plane over the field K can be factored into a finite sequence of quadratic transfor- mations.

Noether’s proof of this theorem is based upon the fact that every bunch of rational curves of the degree n can be simplified to the bunch of rational curves of lower degree by means of quadratic transformations. Therefore, by a finite sequence of quadratic transformations every bunch of rational curves of the degree n can be transform into a bunch of straight lines. Jacob Rosanes5 found this result independently and also proved that every one-to-one algebraic transformation of a plane must be Cremona transformation. Amongst most important properties we should mention that under Cremona transforma- tions neither the degree nor the class of algebraic curves remain invariant in general, but the genus of algebraic curves does. Cremona transformations of plane algebraic curves are still used for the reduction of their singularities. Cremona transformations of three-dimensional space are much more complicated than transformations of the plane. The main difference between plane and space Cremona transfor- mations is that a Cremona transformation of three-dimensional space need not necessarily be of the same degree as its inverse transformation. The most important general theorem concerning Cremona transformations of three-dimensional space is Hudson’s Factorization Theorem:6 There does not exist a finite set of types of Cremona transformations of three- dimensional space of such a nature that any Cremona transformation can be factored into transformations of these types.

3Ernest de Jonqui`eres (1820–1901), a French naval officer and mathematician who discovered many results in geometry. 4Max Noether (1844–1921), a German mathematician, one of the founders of algebraic geometry. 5Jacob Rosanes (1842–1922), a German mathematician who was interested in algebraic geometry and invariant theory; he was also a chess master. 6Hilda Phoebe Hudson (1881–1965), an English mathematician who made significant contributions to Cremona transformations.

36 TRKOVSKA:´ LUIGI CREMONA AND HIS TRANSFORMATIONS

The Influence on the Czech Geometrical School In the second half of the 19th century several mathematicians in the Czech countries were engaged in the problems of birational transformations. The most important one was Emil Weyr (1848–1894) who gained the doctor’s degree in philosophy at the Leipzig University in 1869. In 1870 he was appointed as a private docent of new geometry at the Prague University and next year he received the post of an extraordinary professor of mathematics at the Czech Polytechnic in Prague. In 1875 he was appointed as a full professor at the Vienna University in order to propagate there new geometry successfully developed in Prague. At first, Emil Weyr was interested in one-to-one geometric transformations. In his extensive works from 1869 and 1870 he showed that besides bijective relationships expressing one-to-one projective transformation between two figures there exist also transformations under which more points of one figure correspond to only one point of the second figure. He points out that such transformations are in a close connection with the theory of algebraic curves and surfaces. In the school year 1870/71 Emil Weyr spent a study stay in Milan during which he es- tablished a lifelong friendship and a cooperation with Italian geometers, especially with Luigi Cremona. He took part in his lectures and they discussed various geometric problems. Emil Weyr was the first Czech geometer to have understood the essential significance of Cremona’s geometric work on birational transformations for further development of projective geometry. After his arrival back to Prague he translated two most important Cremona’s books into Czech language.7 Together with his brother Eduard Weyr (1852–1903) he is the author of the first Czech mathematical textbook on projective geometry Z´akladov´evyˇsˇs´ıgeometrie [Foundations of Higher Geometry] which was published in three volumes since 1871. This textbook was writ- ten on the basis of French and German ones, it contains both elementary and special subject matter enriched with their own scientific results.

Conclusion Luigi Cremona had a great influence on the development of projective geometry, he made a significant contribution to the theory of birational transformations. Cremona transformations have been still used for studying rational curves and surfaces. They are effective in the reduction of singularities of plane algebraic curves as well.

Acknowledgments. The author thanks doc. RNDr. Jindˇrich Beˇcv´aˇr,CSc., for his professional help with the dissertation.

References Beˇcv´aˇr,J., Beˇcv´aˇrov´a,M., Skoda,ˇ J., Emil Weyr a jeho pobyt v It´aliiv roce 1870/71, Edition Dˇejiny matematiky, Volume 28, CVUT,ˇ Praha, 2006. Coolidge, J. L., A History of Geometrical Methods, Dover Publications, Mineola, New York, 2003. Ciˇzm´ar,J.,ˇ Biracion´alnetransform´acie1860–1960, historick´yprehl’ad, in Matematika v promˇen´ach vˇek˚uI, Edition Dˇejiny matematiky, Volume 11, Prometheus, Praha, 1998, 79–98. Folta, J., Cesk´ageometrick´aˇskola–ˇ Historick´aanal´yza, Studie Ceskoslovensk´eakademieˇ vˇed,Volume 9, Academia, Praha, 1982. Hudson, H. P., Cremona Transformations in Plane and Space, Cambridge University Press, Cam- bridge, 1927. Weyr, E., Cremonovy geometrick´etransformace ´utvar˚urovinn´ych, Zivaˇ X., n´aklademMusea kr´alovstv´ı Cesk´eho,Praha,ˇ 1872. Weyr, E., Uvod´ do geometrick´etheorie kˇrivekrovinn´ych, Jednota ˇcesk´ych mathematik˚u,Praha, 1873.

7Weyr, E., Cremonovy geometrick´etransformace ´utvar˚urovinn´ych [Cremona’s Geometric Transformations of Plane Figures], 1872, in original Cremona, L., Sulle trasformazioni geometriche delle figure piane, 1863, 1865; Weyr, E., Uvod´ do geometrick´etheorie kˇrivekrovinn´ych [Introduction to Geometric Theory of Plane Curves], 1873, in original Cremona, L., Introduzione ad una teoria geometrica delle curve piane, 1862.

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