Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the Severi type inequalities for irregular surfaces
231
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Let $X$ be a minimal surface of general type and maximal Albanese dimension with irregularity $qgeq 2$. We show that $K_X^2geq 4chi(mathcal O_X)+4(q-2)$ if $K_X^2<frac92chi(mathcal O_X)$, and also obtain the characterization of the equality. As a consequence, we prove a conjecture of Manetti on the geography of irregular surfaces if $K_X^2geq 36(q-2)$ or $chi(mathcal O_X)geq 8(q-2)$, and we also prove a conjecture that surfaces of general type and maximal Albanese dimension with $K_X^2=4chi(mathcal O_X)$ are exactly the resolution of double covers of abelian surfaces branched over ample divisors with at worst simple singularities.
rate research
Read More
Let ${P_i}_{1 leq i leq r}$ and ${Q_i}_{1 leq i leq r}$ be two collections of Brauer Severi surfaces (resp. conics) over a field $k$. We show that the subgroup generated by the $P_is$ in $Br(k)$ is the same as the subgroup generated by the $Q_is$ iff $Pi P_i $ is birational to $Pi Q_i$. Moreover in this case $Pi P_i$ and $Pi Q_i$ represent the same class in $M(k)$, the Grothendieck ring of $k$-varieties. The converse holds if $char(k)=0$. Some of the above implications also hold over a general noetherian base scheme.
For a binary quartic form $phi$ without multiple factors, we classify the quartic K3 surfaces $phi(x,y)=phi(z,t)$ whose Neron-Severi group is (rationally) generated by lines. For generic binary forms $phi$, $psi$ of prime degree without multiple factors, we prove that the Neron-Severi group of the surface $phi(x,y)=psi(z,t)$ is rationally generated by lines.
We introduce the notions of categorical systoles and categorical volumes of Bridgeland stability conditions on triangulated categories. We prove that for any projective K3 surface, there exists a constant C depending only on the rank and discriminant of its Picard group, such that $$mathrm{sys}(sigma)^2leq Ccdotmathrm{vol}(sigma)$$ holds for any stability condition on the derived category of coherent sheaves on the K3 surface. This is an algebro-geometric generalization of a classical systolic inequality on two-tori. We also discuss applications of this inequality in symplectic geometry.
The discriminant, $D$, in the base of a miniversal deformation of an irreducible plane curve singularity, is partitioned according to the genus of the (singular) fibre, or, equivalently, by the sum of the delta invariants of the singular points of the fibre. The members of the partition are known as the {it Severi strata}. The smallest is the $delta$-constant stratum, $D(delta)$, where the genus of the fibre is $0$. It is well known, by work of Givental and Varchenko, to be Lagrangian with respect to the symplectic form $Omega$ obtained by pulling back the intersection form on the cohomology of the fibre via the period mapping. We show that the remaining Severi strata are also co-isotropic with respect to $Omega$, and moreover that the coefficients of the expression of $Omega^{wedge k}$ with respect to a basis of $Omega^{2k}(log D)$ are equations for $D(delta-k+1)$, for $k=1,ldots,delta$. These equations allow us to show that for $E_6$ and $E_8$, $D(delta)$ is Cohen-Macaulay (this was already shown by Givental for $A_{2k}$), and that, as far as we can calculate, for $A_{2k}$ all of the Severi strata are Cohen-Macaulay. Our construction also produces a canonical rank 2 maximal Cohen Macaulay module on the discriminant.
Let $Z$ be a closed subscheme of a smooth complex projective complete intersection variety $Ysubseteq Ps^N$, with $dim Y=2r+1geq 3$. We describe the Neron-Severi group $NS_r(X)$ of a general smooth hypersurface $Xsubset Y$ of sufficiently large degree containing $Z$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
