**Margherita Piazzolla Beloch** (12 July 1879 in Frascati – 28 September 1976 in Rome)^{ [1] } was an Italian mathematician who worked in algebraic geometry, algebraic topology and photogrammetry.

Beloch was the daughter of the German historian Karl Julius Beloch, who taught ancient history for 50 years at Sapienza University of Rome, and American Bella Bailey.^{ [1] }

Beloch studied mathematics at the Sapienza University of Rome and wrote her undergraduate thesis under the supervision of Guido Castelnuovo. She received her degree in 1908^{ [1] } with Lauude and "dignita' di stampa" which means that her work was worthy of publication and in fact her thesis "Sulle trasformazioni birazionali dello spazio" (On Birational Transformations In Space) was published in the Annali di Matematica Pura ed Applicata.^{[ citation needed ]}

Guido Castelnuovo was very impressed with her talent and offer her the position of assistant which Margherita took and held until 1919, when she moved to Pavia and the successive year to Palermo to work under Michele De Franchis, an important figure of the Italian school of algebraic geometry at the time.^{ [1] }

In 1924, Beloch completed her "libera docenza" (a degree that at that time had to be obtained before one could become a professor) and three years later she became a full professor at the University of Ferrara where she taught until her retirement (1955).^{ [1] }

Her main scientific interests were in algebraic geometry, algebraic topology and photogrammetry.

After her thesis, she worked on classification of algebraic surfaces studying the configurations of lines that could lie on surfaces. The next step was to study rational curves lying on surfaces and in this framework Beloch obtained the following important result:^{ [2] } "Hyperelliptic surfaces of rank 2 are characterised by having 16 rational curves."

Beloch also made some contributions to the theory of skew algebraic curves.^{ [3] } She continued working on topological properties of algebraic curves either planar or lying on ruled or cubic surfaces for most of her life, writing about a dozen papers on these subjects.^{ [4] }

Around 1940 Beloch become more and more interested in photogrammetry and the application of mathematics, and in particular algebraic geometry, to it. She is also known for her contribution to the mathematics of paper folding:^{ [5] } In particular she seems to have been the first to formalise an origami move which allows, when possible, to construct by paper folding the common tangents to two parabolas. As a consequence she showed how to extract cubic roots by paper folding,^{ [6] } something that is impossible to do by ruler and compass. The move she used has been called the Beloch fold.^{ [7] }

The art of origami or paper folding has received a considerable amount of mathematical study. Fields of interest include a given paper model's flat-foldability and the use of paper folds to solve mathematical equations.

**Oscar Zariski** was a Russian-born American mathematician and one of the most influential algebraic geometers of the 20th century.

In mathematics, **birational geometry** is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational functions rather than polynomials; the map may fail to be defined where the rational functions have poles.

In relation with the history of mathematics, the **Italian school of algebraic geometry** refers to the work over half a century or more done internationally in birational geometry, particularly on algebraic surfaces. There were in the region of 30 to 40 leading mathematicians who made major contributions, about half of those being in fact Italian. The leadership fell to the group in Rome of Guido Castelnuovo, Federigo Enriques and Francesco Severi, who were involved in some of the deepest discoveries, as well as setting the style.

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**Guido Castelnuovo** was an Italian mathematician. He is best known for his contributions to the field of algebraic geometry, though his contributions to the study of statistics and probability theory are also significant.

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In algebraic geometry, a branch of mathematics, a **rational surface** is a surface birationally equivalent to the projective plane, or in other words a rational variety of dimension two. Rational surfaces are the simplest of the 10 or so classes of surface in the Enriques–Kodaira classification of complex surfaces, and were the first surfaces to be investigated.

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**Guido Zappa** was an Italian mathematician and a noted group theorist: his other main research interests were geometry and also the history of mathematics. Zappa was particularly known for some examples of algebraic curves that strongly influenced the ideas of Francesco Severi.

In algebraic geometry, the **Castelnuovo–Mumford regularity** of a coherent sheaf *F* over projective space **P**^{n} is the smallest integer *r* such that it is **r-regular**, meaning that

**Enzo Martinelli** was an Italian mathematician, working in the theory of functions of several complex variables: he is best known for his work on the theory of integral representations for holomorphic functions of several variables, notably for discovering the Bochner–Martinelli formula in 1938, and for his work in the theory of multi-dimensional residues.

**Giovanni Battista Guccia** was an Italian mathematician.

**Emma Castelnuovo** was an Italian mathematician of Jewish descent.

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**Enrico Bompiani** was an Italian mathematician, specializing in differential geometry.

**Michele de Franchis** was an Italian mathematician, specializing in algebraic geometry. He is known for the De Franchis theorem and the Castelnuovo–de Franchis theorem.

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- 1 2 3 4 5 Kofler, Massimo, "Margherita Beloch Piazzolla",
*Enciclopedia Delle Donne* - ↑ E. Strickland,
*Scienziate d'Italia: diciannove vite per la ricerca*. - ↑ M. Beloch Piazzolla, "Sur le nombre des plurisecantes et sur la classification des courbes gauches algebriques",
*Comptes Rendus de l'Ac. des Sciences*, 1940 - ↑ "Beloch Margherita", at Dept. of Mathematics and Information Science, University of Palermo.
- ↑ Thomas C. Hull, "Solving cubics with creases: the work of Beloch and Lill",
*Amer. Math. Monthly*118 (2011), no. 4, 307–15. - ↑ M. Beloch Piazzolla, "Sul metodo del ripiegamento della carta per la risoluzione dei problemi geometrici",
*Periodico di Mathematiche*Ser. 4, 16 (1936) 104–108. - ↑ Ken Liu, "The Magic and Mathematics of Paper-Folding",
*Tor.com*, June 29, 2017.

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