Divisor

About: Divisor is a research topic. Over the lifetime, 2462 publications have been published within this topic receiving 21394 citations. The topic is also known as: factor & submultiple.

Equivariant quantum cohomology of the odd symplectic Grassmannian

Abstract: The odd symplectic Grassmannian $$\mathrm {IG}:=\mathrm {IG}(k, 2n+1)$$ parametrizes k dimensional subspaces of $${\mathbb {C}}^{2n+1}$$ which are isotropic with respect to a general (necessarily degenerate) symplectic form. The odd symplectic group acts on $$\mathrm {IG}$$ with two orbits, and $$\mathrm {IG}$$ is itself a smooth Schubert variety in the submaximal isotropic Grassmannian $$\mathrm {IG}(k, 2n+2)$$ . We use the technique of curve neighborhoods to prove a Chevalley formula in the equivariant quantum cohomology of $$\mathrm {IG}$$ , i.e. a formula to multiply a Schubert class by the Schubert divisor class. This generalizes a formula of Pech in the case $$k=2$$ , and it gives an algorithm to calculate any multiplication in the equivariant quantum cohomology ring.

Exponential Sums, Cyclic Codes and Sequences: the Odd Characteristic Kasami Case

Abstract: Let $q=p^n$ with $n=2m$ and $p$ be an odd prime. Let $0\leq k\leq n-1$ and $k eq m$. In this paper we determine the value distribution of following exponential(character) sums \[\sum\limits_{x\in \bF_q}\zeta_p^{\Tra_1^m (\alpha x^{p^{m}+1})+\Tra_1^n(\beta x^{p^k+1})}\quad(\alpha\in \bF_{p^m},\beta\in \bF_{q})\] and \[\sum\limits_{x\in \bF_q}\zeta_p^{\Tra_1^m (\alpha x^{p^{m}+1})+\Tra_1^n(\beta x^{p^k+1}+\ga x)}\quad(\alpha\in \bF_{p^m},\beta,\ga\in \bF_{q})\] where $\Tra_1^n: \bF_q\ra \bF_p$ and $\Tra_1^m: \bF_{p^m}\ra\bF_p$ are the canonical trace mappings and $\zeta_p=e^{\frac{2\pi i}{p}}$ is a primitive $p$-th root of unity. As applications: (1). We determine the weight distribution of the cyclic codes $\cC_1$ and $\cC_2$ over $\bF_{p^t}$ with parity-check polynomials $h_2(x)h_3(x)$ and $h_1(x)h_2(x)h_3(x)$ respectively where $t$ is a divisor of $d=\gcd(m,k)$, and $h_1(x)$, $h_2(x)$ and $h_3(x)$ are the minimal polynomials of $\pi^{-1}$, $\pi^{-(p^k+1)}$ and $\pi^{-(p^m+1)}$ over $\bF_{p^t}$ respectively for a primitive element $\pi$ of $\bF_q$. (2). We determine the correlation distribution among a family of m-sequences. This paper extends the results in \cite{Zen Li}.

The Number of Rational Curves on K3 Surfaces

Abstract: Let X be a K3 surface with a primitive ample divisor H, and let $\beta=2[H]\in H_2(X, \mathbf Z)$. We calculate the Gromov-Witten type invariants $n_{\beta}$ by virtue of Euler numbers of some moduli spaces of stable sheaves. Eventually, it verifies Yau-Zaslow formula in the non primitive class $\beta$.

A binary additive equation involving fractional powers

Abstract: Let c be a real number with 1 < c < 2. We study the representations of a large integer n in the form $$ [m^c] + [p^c] = n, $$ where m is an integer and p is a prime number. We prove that when $1 < c < \frac{16}{15}$, all sufficiently large integers are thus representable.

New cubic fourfolds with odd degree unirational parametrizations

Abstract: We prove that the moduli space of cubic fourfolds $\mathcal{C}$ contains a divisor $\mathcal{C}_{42}$ whose general member has a unirational parametrization of degree 13. This result follows from a thorough study of the Hilbert scheme of rational scrolls and an explicit construction of examples. We also show that $\mathcal{C}_{42}$ is uniruled.