Edge operators.
We use floor decompositions of tropical curves to prove that any enumerative problem concerning conics passing through projective-linear subspaces in is maximal. That is, there exist generic configurations of real linear spaces such that all complex conics passing through these constraints are actually real.
We compute the number of irreducible rational curves of given degree with 1 tacnode in or 1 node in meeting an appropriate generic collection of points and lines. As a byproduct, we also compute the number of rational plane curves of degree passing through given points and tangent to a given line. The method is ‘classical’, free of Quantum Cohomology.
First we find effective bounds for the number of dominant rational maps between two fixed smooth projective varieties with ample canonical bundles. The bounds are of the type , where , is the canonical bundle of and are some constants, depending only on .Then we show that for any variety there exist numbers and with the following properties:For any threefold of general type the number of dominant rational maps is bounded above by .The number of threefolds , modulo birational...
We determine the possible even sets of nodes on sextic surfaces in , showing in particular that their cardinalities are exactly the numbers in the set . 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...