This is the first in a sequence of papers in which we construct a quantum version of the Kirwan map from the equivariant quantum cohomology QHG(X) of a smooth polarized complex projective variety X with the action of a connected complex reductive group G to the orbifold quantum cohomology QH(X//G) of its geometric invariant theory quotient X//G, and prove that it intertwines the genus zero gaug...

Let S be the first degeneracy locus of a morphism vector bundles corresponding to general matrix linear forms in $${\mathbb {P}}^s$$ . We prove that, under certain positivity conditions, its Hilbert square $${{\mathrm{Hilb}}}^2(S)$$ is isomorphic zero global section an irreducible homogeneous bundle on product Grassmannians. Our construction involves naturally associated Fano variety, and expli...

The Jacobian Conjecture can be generalized and is established : Let S be a polynomial ring over a field of characteristic zero in finitely may variables. Let T be an unramified, finitely generated extension of S with T = k. Then T = S. Let k be an algebraically closed field, let k be an affine space of dimension n over k and let f : k −→ k be a morphism of algebraic varieties. Then f is given b...

We present in this paper a systematic study on how to morph a well-trained neural network to a new one so that its network function can be completely preserved. We define this as network morphism in this research. After morphing a parent network, the child network is expected to inherit the knowledge from its parent network and also has the potential to continue growing into a more powerful one...

We define tautological Chow classes on the moduli space G d of triples consisting of a curve C, a line bundle L on C of degree d, and a linear system V on L of dimension r. In the case where the forgetful morphism to Mg has relative dimension zero, we describe the images of these classes in A(Mg). As an application, we compute the (virtual) slopes of several different classes of divisors on Mg.

Given a characteristic zero field k and a dominant morphism of affine algebraic k-groups μ : G → C one can form a functor from k-algebras to abelian groups R 7→ F(R) := C(R)/μ(G(R)). Assuming that C is commutative we prove that this functor satisfies a purity theorem for any regular local k-algebra. Few examples are considered in the very end of the preprint.

Abstract We describe all polarizations for abelian varieties over a finite field in fixed isogeny class corresponding to squarefree Weil polynomial, when one variety the admits canonical lifting characteristic zero, that is, which reduction morphism induces an isomorphism of endomorphism rings. Categorical equivalences between fields and fractional ideals étale algebras enable us explicitly com...

In this short note, we will show the following weak evidence of S. Lang conjecture over function elds. Let f : X ! Y be a projective and surjective morphism of algebraic varieties over an algebraically closed eld k of characteristic zero, whose generic ber is geometrically irreducible and of general type. If f is not birationally trivial, then there are countably many proper closed varieties fZ...

(f) Explicitly verify that the the 5 global sections above generate the invertible sheaf OP2(2)|X(−C) at every point of X. Therefore there is a unique morphism φ : X → P such that φOP4(1) equals OP2(2)|X(−C) and the pullback of the homogeneous coordinates are the 5 sections above. (g) Also verify that y0, y1, y2, y3 are identically zero on the line L. Therefore φ contracts L to the point p = [0...