scholarly journals Real Algebraic Curves on Real del Pezzo Surfaces

2020 ◽  
Author(s):  
Matilde Manzaroli
Keyword(s):  
Algebraic Curves ◽  
Topological Type ◽  
Del Pezzo Surfaces ◽  
Toric Surfaces ◽  
Real Algebraic Curves ◽  
Del Pezzo ◽  
Classical Subject ◽  
The 19Th Century ◽  
Type Classification

Abstract The study of the topology of real algebraic varieties dates back to the work of Harnack, Klein, and Hilbert in the 19th century; in particular, the isotopy-type classification of real algebraic curves in real toric surfaces is a classical subject that has undergone considerable evolution. On the other hand, not much is known for more general ambient surfaces. We take a step forward in the study of topological-type classification of real algebraic curves on non-toric surfaces focusing on real del Pezzo surfaces of degree 1 and 2 with multi-components real part. We use degeneration methods and real enumerative geometry in combination with variations of classical methods to give obstructions to the existence of topological-type classes realized by real algebraic curves and to give constructions of real algebraic curves with prescribed topology.

scholarly journals Quotients of del Pezzo surfaces

2019 ◽  
Vol 30 (12) ◽  
pp. 1950068
Author(s):  
Andrey Trepalin
Keyword(s):  
Complete Classification ◽  
Finite Subgroup ◽  
Del Pezzo Surfaces ◽  
Picard Number ◽  
Del Pezzo Surface ◽  
Cyclic Groups ◽  
Trivial Group ◽  
Quotient Surface ◽  
Del Pezzo

Let [Formula: see text] be any field of characteristic zero, [Formula: see text] be a del Pezzo surface and [Formula: see text] be a finite subgroup in [Formula: see text]. In this paper, we study when the quotient surface [Formula: see text] can be non-rational over [Formula: see text]. Obviously, if there are no smooth [Formula: see text]-points on [Formula: see text] then it is not [Formula: see text]-rational. Therefore, under assumption that the set of smooth [Formula: see text]-points on [Formula: see text] is not empty we show that there are few possibilities for non-[Formula: see text]-rational quotients. The quotients of del Pezzo surfaces of degree [Formula: see text] and greater are considered in the author’s previous papers. In this paper, we study the quotients of del Pezzo surfaces of degree [Formula: see text]. We show that they can be non-[Formula: see text]-rational only for the trivial group or cyclic groups of order [Formula: see text], [Formula: see text] and [Formula: see text]. For the trivial group and the group of order [Formula: see text], we show that both [Formula: see text] and [Formula: see text] are not [Formula: see text]-rational if the [Formula: see text]-invariant Picard number of [Formula: see text] is [Formula: see text]. For the groups of order [Formula: see text] and [Formula: see text], we construct examples of both [Formula: see text]-rational and non-[Formula: see text]-rational quotients of both [Formula: see text]-rational and non-[Formula: see text]-rational del Pezzo surfaces of degree [Formula: see text] such that the [Formula: see text]-invariant Picard number of [Formula: see text] is [Formula: see text]. As a result of complete classification of non-[Formula: see text]-rational quotients of del Pezzo surfaces we classify surfaces that are birationally equivalent to quotients of [Formula: see text]-rational surfaces, and obtain some corollaries concerning fields of invariants of [Formula: see text].

scholarly journals Classification of log del Pezzo surfaces of index three

2017 ◽  
Vol 69 (1) ◽  
pp. 163-225 ◽  
Author(s):  
Kento FUJITA ◽  
Kazunori YASUTAKE
Keyword(s):  
Del Pezzo Surfaces ◽  
Del Pezzo ◽  
Log Del Pezzo Surfaces

scholarly journals SIGNATURES OF COLORED LINKS WITH APPLICATION TO REAL ALGEBRAIC CURVES

2005 ◽  
Vol 14 (07) ◽  
pp. 883-918 ◽  
Author(s):  
V. FLORENS
Keyword(s):  
Algebraic Curves ◽  
Betti Numbers ◽  
Alexander Polynomial ◽  
Necessary Condition ◽  
Oriented Link ◽  
Real Algebraic Curves

We construct the signature of a μ-colored oriented link, as a locally constant integer valued function with domain (S1 - {1})μ. It restricts to the Tristram–Levine's signature on the diagonal and the discontinuities can occur only at the zeros of the colored Alexander polynomial. Moreover, the signature and the related nullity verify the Murasugi–Tristram inequality. This gives a new necessary condition for a link to bound a smoothly and properly embedded surface in B4, with given Betti numbers. As an application, we achieve the classification of the complex orientations of maximal plane non-singular projective algebraic curves of degree 7, up to isotopy.

scholarly journals Rigid isotopy classification of real algebraic curves of bidegree (3,3) on quadrics

1999 ◽  
Vol 66 (6) ◽  
pp. 670-674 ◽  
Author(s):  
A. I. Degtyarev ◽  
V. I. Zvonilov
Keyword(s):  
Algebraic Curves ◽  
Real Algebraic Curves

scholarly journals A Characterization of Some Fano 4-folds Through Conic Fibrations

2019 ◽  
Author(s):  
Pedro Montero ◽  
Eleonora Anna Romano
Keyword(s):  
Del Pezzo Surfaces ◽  
Conic Bundle ◽  
Conic Bundles ◽  
Del Pezzo ◽  
Bundle Structure

Abstract We find a characterization for Fano 4-folds $X$ with Lefschetz defect $\delta _{X}=3$: besides the product of two del Pezzo surfaces, they correspond to varieties admitting a conic bundle structure $f\colon X\to Y$ with $\rho _{X}-\rho _{Y}=3$. Moreover, we observe that all of these varieties are rational. We give the list of all possible targets of such contractions. Combining our results with the classification of toric Fano $4$-folds due to Batyrev and Sato we provide explicit examples of Fano conic bundles from toric $4$-folds with $\delta _{X}=3$.

scholarly journals THE HYPERELLIPTIC MAPPING CLASS GROUP OF KLEIN SURFACES

2001 ◽  
Vol 44 (2) ◽  
pp. 351-363 ◽  
Author(s):  
E. Bujalance ◽  
A. F. Costa ◽  
J. M. Gamboa
Keyword(s):  
Moduli Space ◽  
Mapping Class Group ◽  
Class Group ◽  
Algebraic Curves ◽  
Topological Type ◽  
Mapping Class ◽  
Klein Surfaces ◽  
Primary 14H10 ◽  
Generators And Relations ◽  
Real Algebraic Curves

AbstractIn this paper we study the algebraic structure of the hyperelliptic mapping class group of Klein surfaces, which is closely related to the mapping class group of punctured discs. This group plays an important role in the study of the moduli space of hyperelliptic real algebraic curves. Our main result provides a presentation by generators and relations for the hyperelliptic mapping class group of surfaces of prescribed topological type.AMS 2000 Mathematics subject classification: Primary 14H10; 20H10; 30F50

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