We say that an algebraic curve has split jacobian if its jacobian is isogenous to a product of elliptic curves. If X is a curve of genus 2, and /: X —► E a map from X to an elliptic curve, then X has split jacobian. It is not true that a complement to E in the jacobian of X is uniquely determined, but, under certain conditions, there is a canonical choice of elliptic curve E' and algebraic f:X—...

Abstract Involution Schubert polynomials represent cohomology classes of K -orbit closures in the complete flag variety, where is orthogonal or symplectic group. We show they also $\mathsf {T}$ -equivariant subvarieties defined by upper-left rank conditions spaces symmetric skew-symmetric matrices. This geometry implies that these are positive combinations monomials variables $x_i + x_j$ , and ...

We have investigated the role of IL-6 in the initiation and progression of mouse mammary gland involution in IL-6-null mice. This study was based on the hypothesis that IL-6 is the activating cytokine for signal transducer and activator of transcription 3 (Stat), the transcription factor whose presence is required for controlled mammary gland involution. We now show that expression of IL-6 is l...

Let X be a compact connected Riemann surface of genus g, with $$g\, \ge \,2$$ , and let $$\xi $$ holomorphic line bundle on ^{\otimes 2}\,=\, {\mathcal O}_X$$ . Fix theta characteristic $${\mathbb {L}}$$ X. $${\mathcal M}_X(r,\xi )$$ the moduli space stable vector bundles E rank r such that $$\bigwedge ^r E\,=\, \xi $$H^0(X,\, E\otimes {\mathbb L})\,=\, 0$$ Consider quotient by involution given...

We define an invariant of torsors under adjoint linear algebraic groups of type Cn—equivalently, central simple algebras of degree 2n with symplectic involution—for n divisible by 4 that takes values in H(k, μ2). The invariant is distinct from the few known examples of cohomological invariants of torsors under adjoint groups. We also prove that the invariant detects whether a central simple alg...

Involution is the process of post-lactational mammary gland regression to quiescence and it involves secretory epithelial cell death, stroma remodeling and gland repopulation by adipocytes. Though reportedly accompanying apoptosis, the role of autophagy in involution has not yet been determined. We now report that autophagy-related (ATG) proteins mediate dead cell clearance and suppress inflamm...