In algebraic geometry, a **Togliatti surface** is a nodal surface of degree five with 31 nodes. The first examples were constructed by Eugenio G.Togliatti ( 1940 ). ArnaudBeauville ( 1980 ) proved that 31 is the maximum possible number of nodes for a surface of this degree, showing this example to be optimal.

In relation to the history of mathematics, the **Italian school of algebraic geometry** refers to mathematicians and their work in birational geometry, particularly on algebraic surfaces, centered around Rome roughly from 1885 to 1935. There were 30 to 40 leading mathematicians who made major contributions, about half of those being 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 **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

In algebraic geometry, a **surface of general type** is an algebraic surface with Kodaira dimension 2. Because of Chow's theorem any compact complex manifold of dimension 2 and with Kodaira dimension 2 will actually be an algebraic surface, and in some sense most surfaces are in this class.

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, specifically in algebraic geometry and algebraic topology, the **Lefschetz hyperplane theorem** is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties. More precisely, the theorem says that for a variety *X* embedded in projective space and a hyperplane section *Y*, the homology, cohomology, and homotopy groups of *X* determine those of *Y*. A result of this kind was first stated by Solomon Lefschetz for homology groups of complex algebraic varieties. Similar results have since been found for homotopy groups, in positive characteristic, and in other homology and cohomology theories.

In algebraic geometry, a branch of mathematics, a **Hilbert scheme** is a scheme that is the parameter space for the closed subschemes of some projective space, refining the Chow variety. The Hilbert scheme is a disjoint union of projective subschemes corresponding to Hilbert polynomials. The basic theory of Hilbert schemes was developed by Alexander Grothendieck (1961). Hironaka's example shows that non-projective varieties need not have Hilbert schemes.

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.

In mathematics, a **hyperelliptic surface**, or **bi-elliptic surface**, is a surface whose Albanese morphism is an elliptic fibration. Any such surface can be written as the quotient of a product of two elliptic curves by a finite abelian group. Hyperelliptic surfaces form one of the classes of surfaces of Kodaira dimension 0 in the Enriques–Kodaira classification.

In algebraic geometry, a **Barth surface** is one of the complex nodal surfaces in 3 dimensions with large numbers of double points found by Wolf Barth (1996). Two examples are the **Barth sextic** of degree 6 with 65 double points, and the **Barth decic** of degree 10 with 345 double points.

In mathematics, a **fake projective plane** is one of the 50 complex algebraic surfaces that have the same Betti numbers as the projective plane, but are not isomorphic to it. Such objects are always algebraic surfaces of general type.

In mathematics, a **Beauville surface** is one of the surfaces of general type introduced by Arnaud Beauville. They are examples of "fake quadrics", with the same Betti numbers as quadric surfaces.

In algebraic geometry, a **Bordiga surface** is a certain sort of rational surface of degree 6 in *P*^{4}, introduced by Giovanni Bordiga.

**Arnaud Beauville** is a French mathematician, whose research interest is algebraic geometry.

In mathematics, a **Verlinde algebra** is a finite-dimensional associative algebra introduced by Erik Verlinde (1988), with a basis of elements φ_{λ} corresponding to primary fields of a rational two-dimensional conformal field theory, whose structure constants *N*^{ν}_{λμ} describe fusion of primary fields.

In algebraic geometry, an **Endrass surface** is a nodal surface of degree 8 with 168 real nodes, found by Stephan Endrass (1997). As of 2007, it remained the record-holder for the most number of real nodes for its degree; however, the best proven upper bound, 174, does not match the lower bound given by this surface.

In mathematics, the **Labs septic surface** is a degree-7 (septic) nodal surface with 99 nodes found by Labs (2006). As of 2015, it has the largest known number of nodes of a degree-7 surface, though this number is still less than the best known upper bound of 104 nodes given by Varchenko (1983).

In algebraic geometry, a **nodal surface** is a surface in projective space whose only singularities are nodes. A major problem about them is to find the maximum number of nodes of a nodal surface of given degree.

**Dionisio Gallarati** was an Italian mathematician, who specialised in algebraic geometry. He was a major influence on the development of algebra and geometry at the University of Genova.

**Vasilii****Alekseevich Iskovskikh** was a Russian mathematician, specializing in algebraic geometry.

- Beauville, Arnaud (1980), "Sur le nombre maximum de points doubles d'une surface dans ",
*Journées de Géometrie Algébrique d'Angers, Juillet 1979/Algebraic Geometry, Angers, 1979*(PDF) (in French), Alphen aan den Rijn—Germantown, Md.: Sijthoff & Noordhoff, pp. 207–215, MR 0605342 . - Togliatti, Eugenio G. (1940), "Una notevole superficie di 5
^{o}ordine con soli punti doppi isolati",*Beiblatt (Festschrift Rudolf Fueter)*(PDF), Vierteljschr. Naturforsch. Ges. Zürich (in Italian), vol. 85, pp. 127–132, MR 0004492 .

- Endraß, Stephan (2003). "Togliatti surfaces".
- Weisstein, Eric W. "Togliatti surface".
*MathWorld*.

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