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Edge operators.

G. Shabat, V. Shabat, M. Razin (1998)

Mathematica Scandinavica

Effective base point freeness.

János Kollár (1993)

Mathematische Annalen

Eight-secant conics for space curves.

Trygve Johnsen (1992)

Mathematische Zeitschrift

Eine Anzahlbestimmung für mittelbar superoskulierende Kegelschnitte

U. STERZ, K. DRECHSLER (1981)

Beiträge zur Algebra und Geometrie = Contributions to algebra and geometry

Embeddings of general blowing-ups at points.

Marc Coppens (1995)

Journal für die reine und angewandte Mathematik

Enumeration of real conics and maximal configurations

Erwan Brugallé, Nicolas Puignau (2013)

Journal of the European Mathematical Society

We use floor decompositions of tropical curves to prove that any enumerative problem concerning conics passing through projective-linear subspaces in $ℝ P^{n}$ is maximal. That is, there exist generic configurations of real linear spaces such that all complex conics passing through these constraints are actually real.

Enumerative formulae for ruled cubic surfaces and rational quintic curves.

Israel Vainsencher, Daniel F. Coray (1986)

Commentarii mathematici Helvetici

Enumerative geometry of degeneracy loci

Piotr Pragacz (1988)

Annales scientifiques de l'École Normale Supérieure

Enumerative geometry of divisorial families of rational curves

Ziv Ran (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We compute the number of irreducible rational curves of given degree with 1 tacnode in $¶^{2}$ or 1 node in $¶^{3}$ meeting an appropriate generic collection of points and lines. As a byproduct, we also compute the number of rational plane curves of degree $d$ passing through $3 d - 2$ given points and tangent to a given line. The method is ‘classical’, free of Quantum Cohomology.

Enumerative geometry of singular plane curves.

Z. Ran (1989)

Inventiones mathematicae

Estimates of the number of rational mappings from a fixed variety to varieties of general type

Tanya Bandman, Gerd Dethloff (1997)

Annales de l'institut Fourier

First we find effective bounds for the number of dominant rational maps $f : X \to Y$ between two fixed smooth projective varieties with ample canonical bundles. The bounds are of the type ${A \cdot K_{X}^{n}}^{{B \cdot K_{X}^{n}}^{2}}$ , where $n = \dim X$ , $K_{X}$ is the canonical bundle of $X$ and $A, B$ are some constants, depending only on $n$ .Then we show that for any variety $X$ there exist numbers $c (X)$ and $C (X)$ with the following properties:For any threefold $Y$ of general type the number of dominant rational maps $f : X \to Y$ is bounded above by $c (X)$ .The number of threefolds $Y$ , modulo birational...

Even sets of nodes on sextic surfaces

Fabrizio Catanese, Fabio Tonoli (2007)

Journal of the European Mathematical Society

We determine the possible even sets of nodes on sextic surfaces in $ℙ^{3}$ , showing in particular that their cardinalities are exactly the numbers in the set ${24, 32, 40, 56}$ . We also show that all the possible cases admit an explicit description. The methods that we use are an interplay of coding theory and projective geometry on one hand, and of homological and computer algebra on the other. We give a detailed geometric construction for the new case of an even set of 56 nodes, but the ultimate verification of existence...

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