\( % Preview source code from paragraph 0 to 16 %\title[ ]{ } \addtolength{\textwidth}{0mm} \addtolength{\hoffset}{-0mm} \addtolength{\textheight}{0mm} \addtolength{\voffset}{-0mm} %\subjclass{Primary: 14J29} ; Secondary: 14G10, 14G15} \global\long\def\CC{\mathbb{C}}% \global\long\def\BB{\mathbb{B}}% \global\long\def\PP{\mathbb{P}}% \global\long\def\QQ{\mathbb{Q}}% \global\long\def\RR{\mathbb{R}}% \global\long\def\FF{\mathbb{F}}% \global\long\def\LL{\mathbb{L}}% \global\long\def\oo{\mathcal{O}}% \global\long\def\DD{\mathbb{\mathcal{D}}}% \global\long\def\NN{\mathbb{N}}% \global\long\def\ZZ{\mathbb{Z}}% \global\long\def\HH{\mathbb{H}}% \global\long\def\Gal{{\rm Gal}}% \global\long\def\OO{\mathcal{O}}% \global\long\def\pP{\mathfrak{p}}% \global\long\def\pPP{\mathfrak{P}}% \global\long\def\qQ{\mathfrak{q}}% \global\long\def\mm{\mathcal{M}}% \global\long\def\ppp{\mathcal{P}}% \global\long\def\aaa{\mathfrak{a}}% \global\long\def\a{\alpha}% \global\long\def\b{\beta}% \global\long\def\d{\delta}% \global\long\def\D{\Delta}% \global\long\def\L{\Lambda}% \global\long\def\g{\gamma}% \global\long\def\rr{\mathcal{R}}% \global\long\def\oop{\mathcal{O}_{\text{par}}}% \global\long\def\dbf{\mathbf{D}}% \global\long\def\G{\Gamma}% \global\long\def\d{\delta}% \global\long\def\D{\Delta}% \global\long\def\e{\varepsilon}% \global\long\def\k{\kappa}% \global\long\def\l{\lambda}% \global\long\def\m{\mu}% \global\long\def\o{\omega}% \global\long\def\p{\pi}% \global\long\def\P{\Pi}% \global\long\def\s{\sigma}% \global\long\def\S{\Sigma}% \global\long\def\t{\theta}% \global\long\def\T{\Theta}% \global\long\def\f{\varphi}% \global\long\def\deg{{\rm deg}}% \global\long\def\det{{\rm det}}% \global\long\def\Dem{Proof: }% \global\long\def\ker{{\rm Ker}}% \global\long\def\im{{\rm Im}}% \global\long\def\rk{{\rm rk}}% \global\long\def\car{{\rm car}}% \global\long\def\fix{{\rm Fix( }}% \global\long\def\card{{\rm Card }}% \global\long\def\codim{{\rm codim}}% \global\long\def\coker{{\rm Coker}}% \global\long\def\pgcd{{\rm pgcd}}% \global\long\def\ppcm{{\rm ppcm}}% \global\long\def\la{\langle}% \global\long\def\ra{\rangle}% \global\long\def\autp{\text{Aut}_{\text{par}}}% \global\long\def\Alb{{\rm Alb}}% \global\long\def\Jac{{\rm Jac}}% \global\long\def\Disc{{\rm Disc}}% \global\long\def\Tr{{\rm Tr}}% \global\long\def\Nr{{\rm Nr}}% \global\long\def\gast{\Gamma^{\ast}}% \global\long\def\NS{{\rm NS}}% \global\long\def\Pic{{\rm Pic}}% \global\long\def\Km{{\rm Km}}% \global\long\def\rk{{\rm rank}}% \global\long\def\Hom{{\rm Hom}}% \global\long\def\End{{\rm End}}% \global\long\def\aut{{\rm Aut}}% \global\long\def\SSm{{\rm S}}% \global\long\def\psl{{\rm PSL}}% \global\long\def\cu{{\rm (-2)}}% \global\long\def\mod{{\rm \,mod\,}}% \global\long\def\aut{\text{Aut}}% \global\long\def\rls{\mathcal{R}_{\LL\mid S}}% \global\long\def\id{\text{Id}}% \global\long\def\nps{\text{Nef}(X)\cap\mathcal{P}_{S}}% \global\long\def\nef{\text{Nef}(X)}% \global\long\def\rv{R^{\vee}}% \global\long\def\sv{S^{\vee}}% \global\long\def\cmax{c_{\text{max}}}% \global\long\def\ps{\mathcal{P}_{S}}% \global\long\def\prs{\text{pr}_{S}}% \global\long\def\rl{\mathcal{R}_{\LL}}% \global\long\def\prsd{\text{pr}_{S^{\vee}}}% \global\long\def\gcal{\mathcal{G}}% \global\long\def\rrat{\mathcal{R}_{\text{rat}}}% \global\long\def\svs{S_{t}^{\vee}/\,S_{t}}% \global\long\def\auty{\aut(Y)}% \global\long\def\nefyp{\text{Nef}(Y)\cap\mathcal{P}_{\text{NS}(Y)}}% \global\long\def\hbf{\mathbf{H}}% \global\long\def\scal{\mathcal{S}}% \global\long\def\qcal{\mathcal{Q}}% \global\long\def\nx{\mathcal{N}_{X}}% \global\long\def\pl{\mathcal{P}_{\LL}}% \global\long\def\dbp{\;\;}% \global\long\def\btr{\text{\ensuremath{\blacktriangleright}}}% \global\long\def\tr{\triangleright}% \global\long\def\gext{\Gamma_{\text{ext}}}% \global\long\def\NSX{\text{NS}(X)}% \)

2 – Intro | Goals

Study of the $K3$ surfaces $X_t$ with $\NS(X_t)$ isomorphic to the lattice with Gram matrix

$$\begin{pmatrix}2t & 0 & 0\\
0 & -2 & 0\\
0 & 0 & -2
\end{pmatrix}$$

For $1\leq t \leq 50$, we had to determine

  • Automorphism groups $\aut(X_t)$
  • Upper bound on the number of orbits of $\cu$-curves on $X_t$
    under the action of $\aut(X_t)$.