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**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.

**Frascati** is a city and *comune* in the Metropolitan City of Rome Capital in the Lazio region of central Italy. It is located 20 kilometres (12 mi) south-east of Rome, on the Alban Hills close to the ancient city of Tusculum. Frascati is closely associated with science, being the location of several international scientific laboratories.

**Rome** is the capital city and a special *comune* of Italy. Rome also serves as the capital of the Lazio region. With 2,872,800 residents in 1,285 km^{2} (496.1 sq mi), it is also the country's most populated *comune*. It is the fourth most populous city in the European Union by population within city limits. It is the centre of the Metropolitan City of Rome, which has a population of 4,355,725 residents, thus making it the most populous metropolitan city in Italy. Rome is located in the central-western portion of the Italian Peninsula, within Lazio (Latium), along the shores of the Tiber. The Vatican City is an independent country inside the city boundaries of Rome, the only existing example of a country within a city: for this reason Rome has been often defined as capital of two states.

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] }

**Karl Julius Beloch** was a German classical and economic historian.

The **Sapienza University of Rome**, also called simply **Sapienza** or the **University of Rome**, is a collegiate research university located in Rome, Italy. Formally known as **Università degli Studi di Roma** "**La Sapienza**", it is one of the largest European universities by enrollments and one of the oldest in history, founded in 1303. The University is one of the most prestigious Italian universities, commonly ranking first in national rankings and in Southern Europe.

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 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.

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] }

The **University of Ferrara** is the main university of the city of Ferrara in the Emilia-Romagna region of northern Italy. In the years prior to the First World War the University of Ferrara, with more than 500 students, was the best attended of the free universities in Italy. Today there are approximately 16,000 students enrolled at the University of Ferrara with nearly 400 degrees granted each year. The teaching staff number 600, including 288 researchers. It is organized into 12 Departments.

Her main scientific interest were 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] } "Hyperelleptic surfaces of rank 2 are characterised by having 16 rational curves."

**Algebraic geometry** is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.

**Algebraic topology** is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.

**Photogrammetry** falls under the broader category of Geomatics, and, according to the American Society for Photogrammetry and Remote Sensing, is defined as, "the art, science and technology of obtaining reliable information about physical objects and the environment through the process of recording, measuring and interpreting photographic images and patterns of electromagnetic radiant imagery and other phenomena". A simplified definition could be the extraction of three-dimensional measurements from two-dimensional data. Close-range photogrammetry refers to the collection of photography from a lesser distance than traditional aerial photogrammetry. 'Digital' is also an important piece of the name, as this implies the modern digital techniques discussed in this guide. Photogrammetry is as old as modern photography, dating to the mid-19th century and in the simplest example, the distance between two points that lie on a plane parallel to the photographic image plane, can be determined by measuring their distance on the image, if the scale (*s*) of the image is known.

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 rule 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.

In mathematics, a **del Pezzo surface** or **Fano surface** is a two-dimensional Fano variety, in other words a non-singular projective algebraic surface with ample anticanonical divisor class. They are in some sense the opposite of surfaces of general type, which have ample canonical class.

In mathematics, a **rational variety** is an algebraic variety, over a given field *K*, which is birationally equivalent to a projective space of some dimension over *K*. This means that its function field is isomorphic to

**Abramo Giulio Umberto Federigo Enriques** was an Italian mathematician, now known principally as the first to give a classification of algebraic surfaces in birational geometry, and other contributions in algebraic geometry.

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.

In mathematics, **real algebraic geometry** is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them.

**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.

**Pia Maria Nalli** was an Italian mathematician known for her work on the summability of Fourier series, on Morera's theorem for analytic functions of several variables and for finding the solution to the Fredholm integral equation of the third kind for the first time. Her research interests ranged from algebraic geometry to functional analysis and tensor analysis; she was a speaker at the 1928 International Congress of Mathematicians.

**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.

- 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, Thomas C., "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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