arXiv subjects

Frank Sottile

Publications and source records attributed to Frank Sottile.

149 records · Page 9Linked to original sources

arXiv1996

Enumerative geometry for real varieties

We discuss the problem of whether a given problem in enumerative geometry can have all of its solutions be real. In particular, we describe an approach to problems of this type, and show how this can be used to show some enumerative problems involving the Schubert calculus on Grassmannians may have all of their solutions be real. We conclude by describing the work of Fulton and Ronga-Tognoli-Vust, who (independently) showed that there are 5 real plane conics such that each of the 3264 conics tangent to all five are real.

alg-geom↗

arXiv1996

Pieri's Formula Via Explicit Rational Equivalence

We present a new geometric proof of Pieri's formula, exhibiting an explicit chain of rational equivalences from a suitable sum of distinct Schubert varieties to the intersection of a Schubert variety with a special Schubert variety. The geometry of these rational equivalences indicate a link to a combinatorial proof of Pieri's formula using Schensted insertion. This version includes an example to illustrate the proof.

alg-geom↗

arXiv1996

Real enumerative geometry and effective algebraic equivalence

We describe an approach to the question of finding real solutions to problems of enumerative geometry, in particular the question of whether a problem of enumerative geometry can have all of its solutions be real. We give some methods to infer one problem can have all of its solutions be real, given that a related problem does. These are used to show many Schubert-type enumerative problems on some flag manifolds can have all of their solutions be real.

alg-geom↗

arXiv1995

Explicit Enumerative Geometry for the Real Grassmannian of Lines in Projective Space

We extend the classical Schubert calculus of enumerative geometry for the Grassmann variety of lines in projective space from the complex realm to the real. Specifically, given any collection of Schubert conditions on lines in projective space which generically determine a finite number of lines, we show there exist real generic conditions determining the expected number of real lines. Our main tool is an explicit description of rational equivalences which also constitutes a novel determination of the Chow rings of these Grassmann varieties.

alg-geom↗

arXiv1995

Pieri's rule for flag manifolds and Schubert polynomials

We establish the formula for multiplication by the class of a special Schubert variety in the integral cohomology ring of the flag manifold. This formula also describes the multiplication of a Schubert polynomial by either an elementary symmetric polynomial or a complete homogeneous symmetric polynomial. Thus, we generalize the classical Pieri's rule for symmetric polynomials/Grassmann varieties to Schubert polynomials/flag manifolds. Our primary technique is an explicit geometric description of certain intersections of Schubert varieties. This method allows us to compute additional structure constants for the cohomology ring, which we express in terms of paths in the Bruhat order on the symmetric group.

alg-geom↗