Singular Curves of Low Degree and Multifiltrations from Osculating Spaces

In order to study projections of smooth curves, we introduce multifiltrations obtained by combining flags of osculating spaces. We classify all configurations of singularities occurring for a projection of a smooth curve embedded by a complete linear system away from a projective linear space of dimension at most two. In particular, we determine all configurations of singularities of non-degenerate degree $d$ rational curves in $\mathbb{P}^n$ when $d-n\leq 3$ and $d<2n$. Along the way, we describe the Schubert cycles giving rise to these projections. We also reprove a special case of the Castelnuovo bound using these multifiltrations: under the assumption $d<2n$, the arithmetic genus of any non-degenerate degree $d$ curve in $\mathbb{P}^n$ is at most $d-n$.

CRITICAL POINTS OF MASTER FUNCTIONS AND FLAG VARIETIES

We consider critical points of master functions associated with integral dominant weights of Kac–Moody algebras and introduce a generating procedure constructing new critical points starting from a given one. The set of all critical points constructed from a given one is called a population. We formulate a conjecture that a population is isomorphic to the flag variety of the Langlands dual Kac–Moody algebra and prove the conjecture for algebras slN+1, so2N+1, and sp2N. We show that populations associated with a collection of integral dominant slN+1-weights are in one to one correspondence with intersection points of suitable Schubert cycles in a Grassmannian variety.

Pieri’S Formula Via Explicit Rational Equivalence

Pieri’s formula describes the intersection product of a Schubert cycle by a special Schubert cycle on a Grassmannian. We present a new geometric proof, exhibiting an explicit chain of rational equivalences from a suitable sum of distinct Schubert cycles to the intersection of a Schubert cycle with a special Schubert cycle. The geometry of these rational equivalences indicates a link to a combinatorial proof of Pieri’s formula using Schensted insertion.

Schur Q-functions and cohomology of isotropic Grassmannians

We prove that for homogeneous spaces of isotropic Grassmannians the Borel map sends the basis of a truncated algebra of Schur Q-functions consisting of Q-functions or P-functions (depending on a case) onto the basis dual to the basis of Schubert cycles.