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Search: MSC category 14N15 ( Classical problems, Schubert calculus )

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1. CMB Online first

Harada, Megumi; Horiguchi, Tatsuya; Masuda, Mikiya
The equivariant cohomology rings of Peterson varieties in all Lie types
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 efficient 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 $A$.

Keywords:equivariant cohomology, Peterson varieties, flag varieties, Monk formula, Giambelli formula
Categories:55N91, 14N15

2. CMB 2009 (vol 52 pp. 200)

Gatto, Letterio; Santiago, Ta\'\i se
Schubert Calculus on a Grassmann Algebra
The ({\em classical}, {\em small quantum}, {\em equivariant}) cohomology ring of the grassmannian $G(k,n)$ 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 $k$.

Categories:14N15, 14M15

3. CMB 2002 (vol 45 pp. 349)

Coppens, Marc
Very Ample Linear Systems on Blowings-Up at General Points of Projective Spaces
Let $\mathbf{P}^n$ be the $n$-dimensional projective space over some algebraically closed field $k$ of characteristic $0$. For an integer $t\geq 3$ consider the invertible sheaf $O(t)$ on $\mathbf{P}^n$ (Serre twist of the structure sheaf). Let $N = \binom{t+n}{n}$, the dimension of the space of global sections of $O(t)$, and let $k$ be an integer satisfying $0\leq k\leq N - (2n+2)$. Let $P_1,\dots,P_k$ be general points on $\mathbf{P}^n$ and let $\pi \colon X \to \mathbf{P}^n$ be the blowing-up of $\mathbf{P}^n$ at those points. Let $E_i = \pi^{-1} (P_i)$ with $1\leq i\leq k$ be the exceptional divisor. Then $M = \pi^* \bigl( O(t) \bigr) \otimes O_X (-E_1 - \cdots -E_k)$ is a very ample invertible sheaf on $X$.

Keywords:blowing-up, projective space, very ample linear system, embeddings, Veronese map
Categories:14E25, 14N05, 14N15

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