We compute the mixed Hodge structure on rational cohomology of moduli space smooth genus 4 curves. Specifically, we prove that its Poincaré–Serre polynomial is 1 + t2u2 t4u4 t5u6. show this by producing a stratification space, such all strata are geometric quotients complements discriminants.

Abstract We show that the cohomology of perfect cone (also called first Voronoi) toroidal compactification <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi>𝒜</m:mi> <m:mi>g</m:mi> <m:mi>Perf</m:mi> </m:msubsup> </m:math> {{{\mathcal{A}}_{g}^{\operatorname{Perf}}}} moduli space complex principally polarized abelian varieties stabilizes in close to top degree. Moreover, we this stable is purely algebraic, and compute it degree up 13. Our explicit computations...

Conjecture 1.1. Let Xn−1 ⊂ P n be a hypersurface of degree d ≥ and let F(X) G(2, n+ 1) denote the Fano scheme lines on X. B an irreducible component dimension at least − 2. IB := {(x,E) | x ∈ X, E B, PE}, π ρ (respectively) projections to X B. XB = π(IB) ⊆ Cx πρ−1ρπ−1(x). Then, for all ∈XB , ∩Xsing ∅. If we take hyperplane sections in case n, then 1.1 would imply following, which was conjectured independently by Debarre de Jong.

We study families of linear spaces in projective space whose union is a proper subvariety $X$ the expected dimension. establish relations between configurations focal points and existence or non-existence fixed tangent to along general element family. apply our results classification ruled $3$-dimensional varieties.

In this paper we compute the cohomology groups of second Voronoi and perfect cone compactification \mathcal A^{\mathrm{Vor}}_4 A^{\mathrm{perf}}_4 respectively, moduli space abelian fourfolds in degree \leq 9 . The main tool is investigation strata corresponding to semi-abelic varieties with constant torus rank.

We compute the rational cohomology of moduli space non-singular complex plane quartic curves with two marked points. This allows us to calculate projective genus three

In this paper we compute the cohomology groups of second Voronoi compactification moduli space abelian fourfolds in all degrees with exception middle degree 10. We also perfect cone &lt; The main tool is investigation strata corresponding to semi-abelic varieties constant torus rank.

We construct a splitting of the cohomology configuration spaces points on smooth proper variety with multiplicative Chow--K\"unneth decomposition. Applied to hyperelliptic curves, this shows that Torelli group acts trivially rational ordered points. Moreover, if $H_{g,n}$ denotes moduli space $n$-pointed Leray spectral sequence for forgetful map $H_{g,n} \to H_g$ degenerates immediately, in sharp contrast from $M_{g,n}$ $M_g$. This allows new detailed calculations $M_{2,n}$ $n \leq 5$, and...

Abstract We introduce the abstract notion of a smoothable fine compactified Jacobian nodal curve, and family curves whose general element is smooth. Then we combinatorial stability assignment for line bundles their degenerations. prove that Jacobians are in bijection with these assignments. then turn our attention to universal – is, moduli space $\overline {\mathcal {M}}_g$ stable (without marked points). every isomorphic one first constructed by Caporaso, Pandharipande Simpson nineties. In...

We consider projective varieties with degenerate Gauss im age whose focal hypersurfaces are non-reduced schemes.Examples of this situation provided by the secant Severi and Scorza varieties.The moreover characterized a unique ness property.

Abstract We introduce a general abstract notion of fine compactified Jacobian for nodal curves arbitrary genus. focus on genus $1$ and prove combinatorial classification results Jacobians in the case single curve universal family $\overline {{\mathcal {C}}}_{1,n} / \overline {M}}}_{1,n}$ over moduli space stable pointed curves. show that if can be extended to smoothing curve, then it described as sheaves with respect some polarisation. In we construct new examples Jacobians. Then give...

We prove that the Poincare' polynomial of moduli space smooth genus 4 curves is 1+t^2+t^4+t^5. show this by producing a stratification space, such all strata are geometric quotients complements discriminants.