  The very intriguing antagonistic interplay of antiferromagnetism (AF) and
superconductivity (SC), recently discovered in high-temperature
superconductors, is studied in the framework of a microscopic theory. We
explain the surprisingly large increase of the magnetic Bragg peak intensity
$I_{Q}$ at $Q\sim (\pi ,\pi)$ in the magnetic field $H\ll H_{c2}$ at low
temperatures $0<T\ll T_{c},T_{AF}$ in $La_{2-x}Sr_{x}CuO_{4}$. Good agreement
with experimental results is found. The theory predicts large anisotropy of the
relative intensity $R_{Q}(H)=(I_{Q}(H)-I_{Q}(0))/I_{Q}(0)$%, i.e.
$R_{Q}(H\parallel c-axis)\gg R_{Q}(H\perp c-axis)$. The quantum (T=0) phase
diagram at H=0 is constructed. The theory also predicts: (i) the magnetic field
induced AF order in the SC state; (ii) small value for the spin-fluctuation
coupling constant $g<(0.025-0.05)$ $eV$. The latter gives very small SC
critical temperature $T_{c}(\ll 40$ $K)$, thus questioning the spin-fluctuation
mechanism of pairing in HTS oxides.
