Geometry of ℙ2 blown up at seven points

2019 ◽  
Vol 69 (6) ◽  
pp. 1279-1292
Author(s):  
Nabanita Ray
Keyword(s):  
Linear System ◽  
Smooth Surface ◽  
Conic Bundle ◽  
Blowing Up ◽  
System 2 ◽  
Complete Linear System ◽  
Bundle Structure

Abstract In this paper, we prove that blown up at seven general points admits a conic bundle structure over ℙ1 and it can be embedded as (2, 2) divisor in ℙ1 × ℙ2. Conversely, any smooth surface in the complete linear system |(2, 2)| of ℙ1 × ℙ2 can be obtained as an embedding of blowing up ℙ2 at seven points. We also show that smooth surface linearly equivalent to (2, 2) in ℙ1 × ℙ2 has at most four (−2) curves.

ON THE DEGREE AND THE BIRATIONALITY OF THE SECOND ADJUNCTION MAPPING

1999 ◽  
Vol 10 (06) ◽  
pp. 707-719 ◽  
Author(s):  
MAURO C. BELTRAMETTI ◽  
ANDREW J. SOMMESE
Keyword(s):  
Linear System ◽  
Line Bundle ◽  
Mild Condition ◽  
Three Dimensional ◽  
Ample Line Bundle ◽  
Kodaira Dimension ◽  
Projective Manifold ◽  
Very Ample Line Bundle ◽  
System 2 ◽  
Complete Linear System

Let ℒ be a very ample line bundle on ℳ, a projective manifold of dimension n ≥3. Under the assumption that Kℳ + (n-2) ℒ has Kodaira dimension n, we study the degree of the map ϕ associated to the complete linear system |2(KM + (n-2) L)|, where (M, L) is the first reduction of (ℳ, ℒ). In particular we show that under a number of conditions, e.g. n ≥ 5 or Kℳ + (n-3)ℒ having nonnegative Kodaira dimension, the degree of ϕ is one, i.e. ϕ is birational. We also show that under a mild condition on the linear system |KM + (n-2) L| satisfied for all known examples, ϕ is birational unless (ℳ, ℒ) is a three dimensional variety with very restricted invariants. Moreover there is an example with these invariants such that deg ϕ= 2.

scholarly journals On threefolds with low sectional genus

1986 ◽  
Vol 101 ◽  
pp. 27-36 ◽  
Author(s):  
Mauro Beltrametti ◽  
Marino Palleschi
Keyword(s):  
Linear System ◽  
General Problem ◽  
Smooth Surface ◽  
Point Of View ◽  
Ample Divisor ◽  
Sectional Genus ◽  
Complete Linear System

The general problem of rebuilding the threefolds X endowed with a given ample divisor H, possibly non-effective, is closely related to the study of the complete linear system |KX + H| adjoint to H. Many powerful results are known about |KX + H|, for instance when the linear system | H | contains a smooth surface or, more particularly, when H is very ample (e.g. see Sommese [S1] and [S2]). From this point of view we study some properties of |KX + H |, which turn out to be very useful in the description of the threefolds X polarized by an ample divisor H whose arithmetic virtual genus g(H) is sufficiently low.

scholarly journals Curves with Two Pencils and Associated Maps to Projective Spaces

2006 ◽  
Vol 13 (3) ◽  
pp. 411-417
Author(s):  
Edoardo Ballico
Keyword(s):  
Linear System ◽  
Projective Spaces ◽  
Free Resolution ◽  
Homogeneous Ideal ◽  
Projective Curve ◽  
Minimal Free Resolution ◽  
Complete Linear System

Abstract Let 𝑋 be a smooth and connected projective curve. Assume the existence of spanned 𝐿 ∈ Pic𝑎(𝑋), 𝑅 ∈ Pic𝑏(𝑋) such that ℎ0(𝑋, 𝐿) = ℎ0(𝑋, 𝑅) = 2 and the induced map ϕ 𝐿,𝑅 : 𝑋 → 𝐏1 × 𝐏1 is birational onto its image. Here we study the following question. What can be said about the morphisms β : 𝑋 → 𝐏𝑅 induced by a complete linear system |𝐿⊗𝑢⊗𝑅⊗𝑣| for some positive 𝑢, 𝑣? We study the homogeneous ideal and the minimal free resolution of the curve β(𝑋).

scholarly journals The generic isogeny decomposition of the Prym Variety of a cyclic branched covering

2022 ◽  
Author(s):  
Theodosis Alexandrou
Keyword(s):  
Linear System ◽  
Branch Locus ◽  
Prym Variety ◽  
Identity Component ◽  
Branched Covering ◽  
Complete Linear System ◽  
Polarized Surface ◽  
Hyperplane Sections ◽  
General Member ◽  
Projective Surfaces

AbstractLet $$f:S'\longrightarrow S$$ f : S ′ ⟶ S be a cyclic branched covering of smooth projective surfaces over $${\mathbb {C}}$$ C whose branch locus $$\Delta \subset S$$ Δ ⊂ S is a smooth ample divisor. Pick a very ample complete linear system $$|{\mathcal {H}}|$$ | H | on S, such that the polarized surface $$(S, |{\mathcal {H}}|)$$ ( S , | H | ) is not a scroll nor has rational hyperplane sections. For the general member $$[C]\in |{\mathcal {H}}|$$ [ C ] ∈ | H | consider the $$\mu _{n}$$ μ n -equivariant isogeny decomposition of the Prym variety $${{\,\mathrm{Prym}\,}}(C'/C)$$ Prym ( C ′ / C ) of the induced covering $$f:C'{:}{=}f^{-1}(C)\longrightarrow C$$ f : C ′ : = f - 1 ( C ) ⟶ C : $$\begin{aligned} {{\,\mathrm{Prym}\,}}(C'/C)\sim \prod _{d|n,\ d\ne 1}{\mathcal {P}}_{d}(C'/C). \end{aligned}$$ Prym ( C ′ / C ) ∼ ∏ d | n , d ≠ 1 P d ( C ′ / C ) . We show that for the very general member $$[C]\in |{\mathcal {H}}|$$ [ C ] ∈ | H | the isogeny component $${\mathcal {P}}_{d}(C'/C)$$ P d ( C ′ / C ) is $$\mu _{d}$$ μ d -simple with $${{\,\mathrm{End}\,}}_{\mu _{d}}({\mathcal {P}}_{d}(C'/C))\cong {\mathbb {Z}}[\zeta _{d}]$$ End μ d ( P d ( C ′ / C ) ) ≅ Z [ ζ d ] . In addition, for the non-ample case we reformulate the result by considering the identity component of the kernel of the map $${\mathcal {P}}_{d}(C'/C)\subset {{\,\mathrm{Jac}\,}}(C')\longrightarrow {{\,\mathrm{Alb}\,}}(S')$$ P d ( C ′ / C ) ⊂ Jac ( C ′ ) ⟶ Alb ( S ′ ) .

scholarly journals A Characterization of Some Fano 4-folds Through Conic Fibrations

2019 ◽  
Author(s):  
Pedro Montero ◽  
Eleonora Anna Romano
Keyword(s):  
Del Pezzo Surfaces ◽  
Conic Bundle ◽  
Conic Bundles ◽  
Del Pezzo ◽  
Bundle Structure

Abstract We find a characterization for Fano 4-folds $X$ with Lefschetz defect $\delta _{X}=3$: besides the product of two del Pezzo surfaces, they correspond to varieties admitting a conic bundle structure $f\colon X\to Y$ with $\rho _{X}-\rho _{Y}=3$. Moreover, we observe that all of these varieties are rational. We give the list of all possible targets of such contractions. Combining our results with the classification of toric Fano $4$-folds due to Batyrev and Sato we provide explicit examples of Fano conic bundles from toric $4$-folds with $\delta _{X}=3$.

scholarly journals Singular Curves of Low Degree and Multifiltrations from Osculating Spaces

2020 ◽  
Vol 2020 (21) ◽  
pp. 8139-8182 ◽  
Author(s):  
Jarosław Buczyński ◽  
Nathan Ilten ◽  
Emanuele Ventura
Keyword(s):  
Linear System ◽  
Smooth Curve ◽  
Rational Curves ◽  
Arithmetic Genus ◽  
Smooth Curves ◽  
Low Degree ◽  
Complete Linear System ◽  
Osculating Spaces ◽  
Special Case ◽  
Schubert Cycles

Abstract In order to study projections of smooth curves, we introduce multifiltrations obtained by combining flags of osculating spaces. We classify all configurations of singularities occurring for a projection of a smooth curve embedded by a complete linear system away from a projective linear space of dimension at most two. In particular, we determine all configurations of singularities of non-degenerate degree $d$ rational curves in $\mathbb{P}^n$ when $d-n\leq 3$ and $d<2n$. Along the way, we describe the Schubert cycles giving rise to these projections. We also reprove a special case of the Castelnuovo bound using these multifiltrations: under the assumption $d<2n$, the arithmetic genus of any non-degenerate degree $d$ curve in $\mathbb{P}^n$ is at most $d-n$.

scholarly journals ON TWO CONJECTURES FOR CURVES ON K3 SURFACES

2009 ◽  
Vol 20 (12) ◽  
pp. 1547-1560 ◽  
Author(s):  
ANDREAS LEOPOLD KNUTSEN
Keyword(s):  
Linear System ◽  
Additional Condition ◽  
K3 Surfaces ◽  
K3 Surface ◽  
Smooth Curves ◽  
Natural Extensions ◽  
Complete Linear System

We prove that the gonality among the smooth curves in a complete linear system on a K3 surface is constant except for the Donagi–Morrison example. This was proved by Ciliberto and Pareschi under the additional condition that the linear system is ample. The constancy was originally conjectured by Harris and Mumford. As a consequence we prove that exceptional curves on K3 surfaces satisfy the Eisenbud–Lange–Martens–Schreyer conjecture and explicitly describe such curves. They turn out to be natural extensions of the Eisenbud–Lange–Martens–Schreyer examples of exceptional curves on K3 surfaces.

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