We study Schubert polynomials using geometry of infinite-dimensional flag varieties and degeneracy loci. Applications include Graham-positivity of coefficients appearing in equivariant coproduct formulas and expansions of back-stable and enriched Schubert polynomials. We also construct an embedding of the type C flag variety and study the corresponding pullback map on (equivariant) cohomology rings.

In our previous paper, we gave a presentation of the torus-equivariant quantum K-theory ring $QK_{H}(Fl_{n+1})$ of the (full) flag manifold $Fl_{n+1}$ of type $A_{n}$ as a quotient of a polynomial ring by an explicit ideal. In this paper, we prove that quantum double Grothendieck polynomials, introduced by Lenart-Maeno, represent the corresponding (opposite) Schubert classes in the quantum K-theory ring $QK_{H}(Fl_{n+1})$ under this presentation. The main ingredient in our proof is an explicit formula expressing the semi-infinite Schubert class associated to the longest element of the finite Weyl group, which is proved by making use of the general Chevalley formula for the torus-equivariant K-group of the semi-infinite flag manifold associated to $SL_{n+1}(\mathbb {C})$.

In this paper we prove a new generic vanishing theorem for $X$ a complete homogeneous variety with respect to an action of a connected algebraic group. Let $A, B_0\subset X$ be locally closed affine subvarieties, and assume that $B_0$ is smooth and pure dimensional. Let ${\mathcal {P}}$ be a perverse sheaf on $A$ and let $B=g B_0$ be a generic translate of $B_0$. Then our theorem implies $(-1)^{\operatorname {codim} B}\chi (A\cap B, {\mathcal {P}}|_{A\cap B})\geq 0$. As an application, we prove in full generality a positivity conjecture about the signed Euler characteristic of generic triple intersections of Schubert cells. Such Euler characteristics are known to be the structure constants for the multiplication of the Segre–Schwartz–MacPherson classes of these Schubert cells.

Chow rings of flag varieties have bases of Schubert cycles $\sigma _u $, indexed by permutations. A major problem of algebraic combinatorics is to give a positive combinatorial formula for the structure constants of this basis. The celebrated Littlewood–Richardson rules solve this problem for special products $\sigma _u \cdot \sigma _v$, where u and v are p-Grassmannian permutations.

Building on work of Wyser, we introduce backstable clans to prove such a rule for the problem of computing the product $\sigma _u \cdot \sigma _v$ when u is p-inverse Grassmannian and v is q-inverse Grassmannian. By establishing several new families of linear relations among structure constants, we further extend this result to obtain a positive combinatorial rule for $\sigma _u \cdot \sigma _v$ in the case that u is covered in weak Bruhat order by a p-inverse Grassmannian permutation and v is a q-inverse Grassmannian permutation.

We propose two systems of “intrinsic” weights for counting such curves. In both cases the result acquires an exceptionally strong invariance property: it does not depend on the choice of a surface. One of our counts includes all divisor classes of canonical degree 2 and gives in total 30. The other one excludes the class $-2K$, but adds up the results of counting for a pair of real structures that differ by Bertini involution. This count gives 96.

We exhibit basic algebro-geometric results on the formal model of semi-infinite flag varieties and its Schubert varieties over an algebraically closed field ${\mathbb K}$ of characteristic $\neq 2$ from scratch. We show that the formal model of a semi-infinite flag variety admits a unique nice (ind-)scheme structure, its projective coordinate ring has a $\mathbb {Z}$-model and it admits a Frobenius splitting compatible with the boundaries and opposite cells in positive characteristic. This establishes the normality of the Schubert varieties of the quasi-map space with a fixed degree (instead of their limits proved in [K, Math. Ann. 371 no.2 (2018)]) when $\mathsf {char}\, {\mathbb K} =0$ or $\gg 0$, and the higher-cohomology vanishing of their nef line bundles in arbitrary characteristic $\neq 2$. Some particular cases of these results play crucial roles in our proof [47] of a conjecture by Lam, Li, Mihalcea and Shimozono [60] that describes an isomorphism between affine and quantum K-groups of a flag manifold.

We prove an explicit inverse Chevalley formula in the equivariant K-theory of semi-infinite flag manifolds of simply laced type. By an ‘inverse Chevalley formula’ we mean a formula for the product of an equivariant scalar with a Schubert class, expressed as a $\mathbb {Z}\left [q^{\pm 1}\right ]$ -linear combination of Schubert classes twisted by equivariant line bundles. Our formula applies to arbitrary Schubert classes in semi-infinite flag manifolds of simply laced type and equivariant scalars $e^{\lambda }$ , where $\lambda $ is an arbitrary minuscule weight. By a result of Stembridge, our formula completely determines the inverse Chevalley formula for arbitrary weights in simply laced type except for type $E_8$ . The combinatorics of our formula is governed by the quantum Bruhat graph, and the proof is based on a limit from the double affine Hecke algebra. Thus our formula also provides an explicit determination of all nonsymmetric q-Toda operators for minuscule weights in ADE type.

We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field $k$, generalizing the counts that over ${\mathbf {C}}$ there are $27$ lines, and over ${\mathbf {R}}$ the number of hyperbolic lines minus the number of elliptic lines is $3$. In general, the lines are defined over a field extension $L$ and have an associated arithmetic type $\alpha$ in $L^*/(L^*)^2$. There is an equality in the Grothendieck–Witt group $\operatorname {GW}(k)$ of $k$,

\[ \sum_{\text{lines}} \operatorname{Tr}_{L/k} \langle \alpha \rangle = 15 \cdot \langle 1 \rangle + 12 \cdot \langle -1 \rangle, \]

$\operatorname {Tr}_{L/k}$

$\operatorname {GW}(L) \to \operatorname {GW}(k)$

${\mathbf {C}}$

${\mathbf {R}}$

$\mathbf {A}^1$

We previously obtained a congruence modulo four for the number of real solutions to many Schubert problems on a square Grassmannian given by osculating flags. Here we consider Schubert problems given by more general isotropic flags, and prove this congruence modulo four for the largest class of Schubert problems that could be expected to exhibit this congruence.

We consider Schubert problems with respect to flags osculating the rational normal curve. These problems are of special interest when the osculation points are all real. In this case, for zero-dimensional Schubert problems, the solutions are “ as real as possible”. Recent work by Speyer has extended the theory to the moduli space $\overline{{{\mathcal{M}}_{0,\,r}}}$ allowing the points to collide. This gives rise to smooth covers $\overline{{{\mathcal{M}}_{0,\,r}}}\left( \mathbb{R} \right)$, with structure and monodromy described by Young tableaux and jeu de taquin.

In this paper, we give analogous results on one-dimensional Schubert problems over $\overline{{{\mathcal{M}}_{0,\,r}}}$.Their(real) geometry turns out to be described by orbits of Schützenberger promotion and a related operation involving tableau evacuation. Over ${{\mathcal{M}}_{0,\,r}}$, our results show that the real points of the solution curves are smooth.

We also find a new identity involving “first-order” $\text{K}$-theoretic Littlewood-Richardson coefficients, for which there does not appear to be a known combinatorial proof.

We show that the product rank of the $3\,\times \,3$ determinant ${{\det }_{3}}$ is $5$ , and the product rank of the $3\,\times \,3$ permanent $\text{per}{{\text{m}}_{3}}$ is $4$ . As a corollary, we obtain that the tensor rank of ${{\det }_{3}}$ is $5$ and the tensor rank of $\text{per}{{\text{m}}_{3}}$ is $4$ . We show moreover that the border product rank of $\text{per}{{\text{m}}_{3}}$ is larger than $n$ for any $n\,\ge \,3$ .

Let $G$ be a complex semisimple linear algebraic group and let Pet be the associated Peterson variety in the flag variety $G/B$. The main theorem of this note gives an eõcient presentation of the equivariant cohomology ring $H_{S}^{*}$ (Pet) of the Peterson variety as a quotient of a polynomial ring by an ideal $J$ generated by quadratic polynomials, in the spirit of the Borel presentation of the cohomology of the flag variety. Here the group $S\,\cong \,{{\mathbb{C}}^{*}}$ is a certain subgroup of a maximal torus $T$ of $G$. Our description of the ideal $J$ uses the Cartan matrix and is uniform across Lie types. In our arguments we use the Monk formula and Giambelli formula for the equivariant cohomology rings of Peterson varieties for all Lie types, as obtained in the work of Drellich. Our result generalizes a previous theorem of Fukukawa–Harada–Masuda, which was only for Lie type $A$.

Using the Stückrad–Vogel self-intersection cycle of an irreducible and reduced curve in pro-jective space, we obtain a formula that relates the degree of the secant variety, the degree and the genus of the curve and the self-intersection numbers, the multiplicities and the number of branches of the curve at its singular points. From this formula we deduce an expression for the difference between the genera of the curve. This result shows that the self-intersection multiplicity of a curve in projective N-space at a singular point is a natural generalization of the intersection multiplicity of a plane curve with its generic polar curve. In this approach, the degree of the secant variety (up to a factor 2), the self-intersection numbers and the multiplicities of the singular points are leading coefficients of a bivariate Hilbert polynomial, which can be computed by computer algebra systems.

We prove that $r$ independent homogeneous polynomials of the same degree $d$ become dependent when restricted to any hyperplane if and only if their inverse system parameterizes a variety whose $(d-1)$- osculating spaces have dimension smaller than expected. This gives an equivalence between an algebraic notion (called the Weak Lefschetz Property) and a differential geometric notion, concerning varieties that satisfy certain Laplace equations. In the toric case, some relevant examples are classified, and as a byproduct we provide counterexamples to Ilardi's conjecture.

We give bounds for the Betti numbers of projective algebraic varieties in terms of their classes (degrees of dual varieties of successive hyperplane sections). We also give bounds for classes in terms of ramification volumes (mixed ramification degrees), sectional genus and, eventually, in terms of dimension, codimension and degree. For varieties whose degree is large with respect to codimension, we give sharp bounds for the above invariants and classify the varieties on the boundary, thus obtaining a generalization of Castelnuovo’s theory for curves to varieties of higher dimension.

The (classical, small quantum, equivariant) cohomology ring of the grassmannian $G\left( k,\,n \right)$ is generated by certain derivations operating on an exterior algebra of a free module of rank $n$ (Schubert calculus on a Grassmann algebra). Our main result gives, in a unified way, a presentation of all such cohomology rings in terms of generators and relations. Using results of Laksov and Thorup, it also provides a presentation of the universal factorization algebra of a monic polynomial of degree $n$ into the product of two monic polynomials, one of degree $k$.

We study the regularity of the higher secant varieties of ${{\mathbb{P}}^{1}}\times {{\mathbb{P}}^{n}}$, embedded with divisors of type $\text{(}d\text{,}\,\text{2)}$ and $(d,3)$. We produce, for the highest defective cases, a “determinantal” equation of the secant variety. As a corollary, we prove that the Veronese triple embedding of ${{\mathbb{P}}^{n}}$ is not Grassmann defective.

We consider the $k$-osculating varieties ${{O}_{k,\,n.d}}$ to the (Veronese) $d$-uple embeddings of ${{\mathbb{P}}^{n}}$. We study the dimension of their higher secant varieties via inverse systems (apolarity). By associating certain 0-dimensional schemes $Y\,\subset \,{{\mathbb{P}}^{n}}$ to $O_{k,n,d}^{s}$ and by studying their Hilbert functions, we are able, in several cases, to determine whether those secant varieties are defective or not.

This is a note on the classical Waring's problem for algebraic forms. Fix integers $(n,d,r,s)$, and let $\Lambda $ be a general $r$-dimensional subspace of degree $d$ homogeneous polynomials in $n+1$ variables. Let $\mathcal{A}$ denote the variety of $s$-sided polar polyhedra of $\Lambda $. We carry out a case-by-case study of the structure of $\mathcal{A}$ for several specific values of $(n,d,r,s)$. In the first batch of examples, $\mathcal{A}$ is shown to be a rational variety. In the second batch, $\mathcal{A}$ is a finite set of which we calculate the cardinality.

We explain how A. Givental’s mirror symmetric family [14] to the type A ag variety and its proposed generalization [3] to partial ag varieties by Batyrev, Ciocan-Fontanine, Kim and van Straten relate to the Peterson variety Y ⊂ SLn/B. We then use this theory to describe the totally nonnegative part of Y, extending a result from [30].