A conjecture of Serre predicts a precise from of Torelli Theorem for genus 3 curves, namely, an indecomposable principally polarized abelian threefold is a Jacobian if and only if some specific invariant is a square. We study here a three dimensional family of such threefolds, introduced by Howe, Leprevost and Poonen. By a new formulation, we link their results to the conjecture of Serre. Then,...

We study admissible subcategories of derived categories coherent sheaves on del Pezzo surfaces and rational elliptic surfaces. Using a relation between anticanonical divisors we prove the following results. First, classify all projective plane by showing that each is generated subcollection full exceptional collection. Second, show do not contain any phantom subcategories. This provides first e...

Let $A$ be a right coherent ring and $\mathcal{X}$ contravariantly finite subcategory of ${\rm{mod}}\mbox{-}A$ containing projectives. In this paper, we construct recollement abelian categories $({\rm{mod}}_{0}\mbox{-}\mathcal{X}, {\rm{mod}}\mbox{-}\mathcal{X}, {\rm{mod}}\mbox{-}A)$, where ${\rm{mod}}_{0}\mbox{-}\mathcal{X}$ is full ${\rm{mod}}\mbox{-}\mathcal{X}$ consisting all functors vanish...

the unique normalized cusp form of weight 12 with respect to the full modular group. Although Lehmer’s speculation that τ(n) 6= 0 for every positive n remains open, Serre [S] has made substantial progress on the basic question regarding the number of Fourier coefficients of a modular form which can be zero. He shows (see [p. 179, S]) that τ(n) is non-zero for the vast majority of n. In the same...