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.

Donaldson-Thomas theory of A_n x P^1

Abstract: We study the relative Donaldson-Thomas theory of A_n x P^1, where A_n is the surface resolution of a type A_n singularity. The action of divisor operators in the theory is expressed in terms of operators of the affine algebra \hat{gl}(n+1) on Fock space. Assuming a nondegeneracy conjecture, this gives a complete solution for the theory. The results complete the comparison of this theory with the Gromov-Witten theory of A_n x P^1 and the quantum cohomology of the Hilbert scheme of points on A_n.

Fulton's Conjecture for \bar{M}_{0,7}

Abstract: Fulton's conjecture for the moduli space of stable pointed rational curves, \bar{M}_{0,n}, claims that a divisor non-negatively intersecting all F-curves is linearly equivalent to an effective sum of boundary divisors. Our main result is a proof of Fulton's conjecture for n=7. A key ingredient in the proof is an \binom{n}{4} dimensional-subspace of the Neron-Severi space of \bar{M}_{0,n}, defined by averages of Keel relations, for which we prove Fulton's conjecture for all n.

A Class of Sparse Invertible Matrices and Their Use for Nonlinear Prediction of Nearly Periodic Time Series with Fixed Period

Abstract: We introduce a class of sparse matrices U m (A p 1 ) of order m by m, where m is a composite natural number, p 1 is a divisor of m, and A p 1 is a set of nonzero real numbers of length p 1. The construction of U m (A p 1 ) is achieved by iteration, involving repetitive dilation operations and block-matrix operations. We prove that the matrices U m (A p 1 ) are invertible and we compute the inverse matrix (U m (A p 1 ))−1 explicitly. We prove that each row of the inverse matrix (U m (A p 1 ))−1 has only two nonzero entries with alternative signs, located at specific positions, related to the divisors of m. We use the structural properties of the matrix (U m (A p 1 ))−1 in order to build a nonlinear estimator for prediction of nearly periodic time series of length m with fixed period.

A necessary and sufficient condition for Riemann's singularity theorem to hold on a Prym theta divisor

Abstract: Let $(P,\Xi)$ be the naturally polarized model of the Prym variety associated to the etale double cover $\pi : \tilde C\rightarrow C$ of smooth connected curves defined over an algebraically closed field k of characteristic $ e 2$ , where genus( C ) = $g \ge 3$ , Pic $^{(2g-2)}(\tilde C) \supset P = \{\mathcal L \in {\rm Pic}^{(2g-2)}(\tilde C) : {\rm Nm}(\mathcal L) = \omega_C$ and $h^0(\tilde C,\mathcal L)$ is even\} is the Prym variety, and $P \supset \Xi = \{\mathcal L \in P: h^0(\tilde C,\mathcal L) >0 \}$ is the Prym theta divisor with its reduced scheme structure. If $\mathcal L$ is any point on $\Xi$ , we prove that ‘Riemann's singularity theorem holds at $\mathcal L$ ’, i.e. mult $_{\mathcal L}(\Xi) = (1/2)h^0(\tilde C,\mathcal L)$ , if and only if $\mathcal L$ cannot be expressed as $\pi^*(\mathcal M)(B)$ where $B \ge 0$ is an effective divisor on $\tilde C$ , and $\mathcal M$ is a line bundle on C with $h^0(C,\mathcal M) >(1/2)h^0(\tilde C,\mathcal L)$ . This completely characterizes points of $\Xi$ where the tangent cone is the set theoretic restriction of the tangent cone of $\tilde {\Theta}$ , hence also those points on $\Xi$ where Mumford's Pfaffian equation defines the tangent cone to $\Xi$ .

On solutions of the Schlesinger equation in the neigborhood of the Malgrange Θ-divisor

Abstract: We say that the family (1) is isomonodromic if, for all a ∈ D(a0), the monodromies χa : π1(C \ {a1, . . . , an}) → G = GL(p,C) of the corresponding system are equal to each other. (Under small variations of the parameter a, there exists a canonical isomorphism of the fundamental groups π1(C \ {a1, . . . , an}) and π1(C \ {a1, . . . , an}) generating the canonical isomorphism Hom(π1(C \ {a1, . . . , an}), G)/G ∼= Hom(π1(C \ {a1, . . . , an}), G)/G of the spaces of classes of the duality representations for these fundamental groups; this allows one to compare χa for various a ∈ D(a0).) For example, if the matrix Bi(a) satisfies the Schlesinger equation dBi(a) = − n ∑