scholarly journals SEVERI VARIETIES AND BRANCH CURVES OF ABELIAN SURFACES OF TYPE (1, 3)

2002 ◽  
Vol 13 (03) ◽  
pp. 227-244 ◽  
Author(s):  
H. LANGE ◽  
E. SERNESI
Keyword(s):  
Linear System ◽  
Plane Curve ◽  
Main Idea ◽  
Abelian Surface ◽  
Severi Varieties ◽  
Severi Variety ◽  
System L ◽  
Discriminant Curve ◽  
Branch Curve

A polarized abelian surface (A, L) of type (1, 3) induces a 6:1 covering of A onto the projective plane with branch curve, a plane curve B of degree 18. The main result of the paper is that for a general abelian surface of type (1, 3), the curve B is irreducible and reduced and admits 72 cusps, 36 nodes or tacnodes, each tacnode counting as 2 nodes, 72 flexes and 36 bitangents. The main idea of the proof is that for a general (A, L) the discriminant curve in the linear system |L| coincides with the closure of the Severi variety of curves in |L| admitting a node and is dual to the curve B in the sense of projective geometry.

scholarly journals Severi varieties and Brill–Noether theory of curves on abelian surfaces

2019 ◽  
Vol 2019 (749) ◽  
pp. 161-200 ◽  
Author(s):  
Andreas Leopold Knutsen ◽  
Margherita Lelli-Chiesa ◽  
Giovanni Mongardi
Keyword(s):  
Hyperplane Section ◽  
Abelian Surface ◽  
Moduli Space Of Curves ◽  
Arithmetic Genus ◽  
Abelian Surfaces ◽  
Nodal Curves ◽  
Smooth Curves ◽  
Severi Varieties ◽  
Severi Variety ◽  
Nodal Model

Abstract Severi varieties and Brill–Noether theory of curves on K3 surfaces are well understood. Yet, quite little is known for curves on abelian surfaces. Given a general abelian surface S with polarization L of type {(1,n)} , we prove nonemptiness and regularity of the Severi variety parametrizing δ-nodal curves in the linear system {|L|} for {0\leq\delta\leq n-1=p-2} (here p is the arithmetic genus of any curve in {|L|} ). We also show that a general genus g curve having as nodal model a hyperplane section of some {(1,n)} -polarized abelian surface admits only finitely many such models up to translation; moreover, any such model lies on finitely many {(1,n)} -polarized abelian surfaces. Under certain assumptions, a conjecture of Dedieu and Sernesi is proved concerning the possibility of deforming a genus g curve in S equigenerically to a nodal curve. The rest of the paper deals with the Brill–Noether theory of curves in {|L|} . It turns out that a general curve in {|L|} is Brill–Noether general. However, as soon as the Brill–Noether number is negative and some other inequalities are satisfied, the locus {|L|^{r}_{d}} of smooth curves in {|L|} possessing a {g^{r}_{d}} is nonempty and has a component of the expected dimension. As an application, we obtain the existence of a component of the Brill–Noether locus {{\mathcal{M}}^{r}_{p,d}} having the expected codimension in the moduli space of curves {{\mathcal{M}}_{p}} . For {r=1} , the results are generalized to nodal curves.

scholarly journals Peroxiredoxin 6 Attenuates Alloxan-Induced Type 1 Diabetes Mellitus in Mice and Cytokine-Induced Cytotoxicity in RIN-m5F Beta Cells

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Elena G. Novoselova ◽  
Olga V. Glushkova ◽  
Sergey M. Lunin ◽  
Maxim O. Khrenov ◽  
Svetlana B. Parfenyuk ◽  
...  
Keyword(s):  
Type 1 Diabetes ◽  
Beta Cells ◽  
Cell Apoptosis ◽  
Pancreatic Beta Cells ◽  
Apoptotic Cell ◽  
Main Idea ◽  
Severe Form ◽  
Peroxiredoxin 6 ◽  
Cells Cultured

Type 1 diabetes is associated with the destruction of pancreatic beta cells, which is mediated via an autoimmune mechanism and consequent inflammatory processes. In this article, we describe a beneficial effect of peroxiredoxin 6 (PRDX6) in a type 1 diabetes mouse model. The main idea of this study was based on the well-known data that oxidative stress plays an important role in pathogenesis of diabetes and its associated complications. We hypothesised that PRDX6, which is well known for its various biological functions, including antioxidant activity, may provide an antidiabetic effect. It was shown that PRDX6 prevented hyperglycemia, lowered the mortality rate, restored the plasma cytokine profile, reversed the splenic cell apoptosis, and reduced the β cell destruction in Langerhans islets in mice with a severe form of alloxan-induced diabetes. In addition, PRDX6 protected rat insulinoma RIN-m5F β cells, cultured with TNF-α and IL-1β, against the cytokine-induced cytotoxicity and reduced the apoptotic cell death and production of ROS. Signal transduction studies showed that PRDX6 prevented the activation of NF-κB and c-Jun N-terminal kinase signaling cascades in RIN-m5F β cells cultured with cytokines. In conclusion, there is a prospect for therapeutic application of PRDX6 to delay or even prevent β cell apoptosis in type 1 diabetes.

scholarly journals Effective Faithful Tropicalizations Associated to Adjoint Linear Systems

2018 ◽  
Vol 2019 (19) ◽  
pp. 6089-6112
Author(s):  
Shu Kawaguchi ◽  
Kazuhiko Yamaki
Keyword(s):  
Linear System ◽  
Line Bundle ◽  
Projective Variety ◽  
Ample Line Bundle ◽  
Smooth Projective Variety ◽  
Discrete Valuation Ring ◽  
Complete Discrete Valuation Ring ◽  
System L ◽  
Semistable Model ◽  
Adjoint Bundle

Abstract Let R be a complete discrete valuation ring of equi-characteristic zero with fraction field K. Let X be a connected smooth projective variety of dimension d over K, and let L be an ample line bundle over X. We assume that there exist a regular strictly semistable model ${\mathscr {X}}$ of X over R and a relatively ample line bundle ${\mathscr {L}}$ over ${\mathscr {X}}$ with $\left .{{\mathscr {L}}}\right \vert_{{X}} \cong L$. Let $S({\mathscr {X}})$ be the skeleton associated to ${\mathscr {X}}$ in the Berkovich analytification Xan of X. In this article, we study when $S({\mathscr {X}})$ is faithfully tropicalized into tropical projective space by the adjoint linear system |L⊗m ⊗ ωX|. Roughly speaking, our results show that if m is an integer such that the adjoint bundle is basepoint free, then the adjoint linear system admits a faithful tropicalization of $S({\mathscr {X}})$.

scholarly journals Constructions of Brauer-Severi Varieties and Norm Hypersurfaces

1990 ◽  
Vol 42 (2) ◽  
pp. 230-238 ◽  
Author(s):  
Ming-Chang Kang
Keyword(s):  
Function Field ◽  
Splitting Field ◽  
Severi Varieties ◽  
Severi Variety ◽  
Central Simple

Let k be any field, A a central simple k-algebra of degree m (i.e., dimk A = m2). Several methods of constructing the generic splitting fields for A are proposed and Saltman proves that these methods result in almost the same generic splitting field [8, Theorems 4.2 and 4.4]. In fact, the generic splitting field constructed by Roquette [7] is the function field of the Brauer- Severi variety Vm(A) while the generic splitting field constructed by Heuser and Saltman [4 and 8] is the function field of the norm surface W(A). In this paper, to avoid possible confusion about the dimension, we shall call it the norm hypersurface instead of the norm surface.

An algorithm for a lifted Massey triple product of a smooth projective plane curve

2020 ◽  
Vol 30 (08) ◽  
pp. 1651-1669
Author(s):  
Younggi Lee ◽  
Jeehoon Park ◽  
Junyeong Park ◽  
Jaehyun Yim
Keyword(s):  
Projective Plane ◽  
Homogeneous Polynomial ◽  
Plane Curve ◽  
Main Idea ◽  
Triple Product ◽  
Cup Product ◽  
Smooth Projective Plane ◽  
Defining System ◽  
Polynomial Formula ◽  
Massey Triple Product

We provide an explicit algorithm to compute a lifted Massey triple product relative to a defining system for a smooth projective plane curve [Formula: see text] defined by a homogeneous polynomial [Formula: see text] over a field. The main idea is to use the description (due to Carlson and Griffiths) of the cup product for [Formula: see text] in terms of the multiplications inside the Jacobian ring of [Formula: see text] and the Cech–deRham complex of [Formula: see text]. Our algorithm gives a criterion whether a lifted Massey triple product vanishes or not in [Formula: see text] under a particular nontrivial defining system of the Massey triple product and thus can be viewed as a generalization of the vanishing criterion of the cup product in [Formula: see text] of Carlson and Griffiths. Based on our algorithm, we provide explicit numerical examples by running the computer program.

scholarly journals Birational classification of curves on rational surfaces

2010 ◽  
Vol 199 ◽  
pp. 43-93
Author(s):  
Alberto Calabri ◽  
Ciro Ciliberto
Keyword(s):  
Linear System ◽  
Plane Curve ◽  
Rational Surface ◽  
Minimal Degree ◽  
Classification Theorem ◽  
Plane Curves ◽  
Cremona Transformation ◽  
Rational Surfaces

AbstractIn this paper we consider the birational classification of pairs (S, ℒ), withSa rational surface andℒa linear system onS. We give a classification theorem for such pairs, and we determine, for each irreducible plane curveB, itsCremona minimalmodels, that is, those plane curves which are equivalent toBvia a Cremona transformation and have minimal degree under this condition.

scholarly journals A Note on Planarity Stratification of Hurwitz Spaces

2015 ◽  
Vol 58 (3) ◽  
pp. 596-609 ◽  
Author(s):  
Jared Ongaro ◽  
Boris Shapiro
Keyword(s):  
Riemann Surface ◽  
Meromorphic Function ◽  
Plane Curve ◽  
Minimal Degree ◽  
Hurwitz Spaces ◽  
Birational Map ◽  
Severi Varieties ◽  
Closed Riemann Surface

AbstractOne can easily show that any meromorphic function on a complex closed Riemann surface can be represented as a composition of a birational map of this surface to ℂℙ2and a projection of the image curve froman appropriate pointp∊ ℂℙ2to the pencil of lines throughp. We introduce a natural stratiûcation of Hurwitz spaces according to the minimal degree of a plane curve such that a given meromorphic function can be represented in the above way and calculate the dimensions of these strata. We observe that they are closely related to a family of Severi varieties studied earlier by J. Harris, Z. Ran, and I. Tyomkin.

ON RAMIFIED COVERS OF THE PROJECTIVE PLANE I: INTERPRETING SEGRE'S THEORY (WITH AN APPENDIX BY EUGENII SHUSTIN)

2011 ◽  
Vol 22 (05) ◽  
pp. 619-653 ◽  
Author(s):  
MICHAEL FRIEDMAN ◽  
MAXIM LEYENSON ◽  
EUGENII SHUSTIN
Keyword(s):  
Projective Plane ◽  
Smooth Surface ◽  
Plane Curve ◽  
Main Question ◽  
Generic Projection ◽  
Complete Answer ◽  
Classical Work ◽  
Numerical Invariants ◽  
Ramified Covers ◽  
Branch Curve

We study ramified covers of the projective plane ℙ2. Given a smooth surface S in ℙN and a generic enough projection ℙN → ℙ2, we get a cover π: S → ℙ2, which is ramified over a plane curve B. The curve B is usually singular, but is classically known to have only cusps and nodes as singularities for a generic projection. The main question that arises is with respect to the geometry of branch curves; i.e. how can one distinguish a branch curve from a non-branch curve with the same numerical invariants? For example, a plane sextic with six cusps is known to be a branch curve of a generic projection iff its six cusps lie on a conic curve, i.e. form a special 0-cycle on the plane. The classical work of Beniamino Segre gives a complete answer to the second question in the case when S is a smooth surface in ℙ3. We give an interpretation of the work of Segre in terms of relation between Picard and Chow groups of 0-cycles on a singular plane curve B. In addition, the appendix written by E. Shustin shows the existence of new Zariski pairs.

scholarly journals LINE BUNDLES OF TYPE (1,…,1,2,…,2,4,…,4) ON ABELIAN VARIETIES

2001 ◽  
Vol 12 (01) ◽  
pp. 125-142
Author(s):  
JAYA N. IYER
Keyword(s):  
Linear System ◽  
Abelian Variety ◽  
Moduli Space ◽  
Abelian Varieties ◽  
Line Bundles ◽  
Complete Linear System

We show birationality of the morphism associated to line bundles L of type (1,…,1,2,…,2,4,…,4) on a generic g-dimensional abelian variety into its complete linear system such that h0(L) = 2g. When g = 3, we describe the image of the abelian threefold and from the geometry of the moduli space SUC(2) in the linear system |2θC|, we obtain analogous results in ℙH0(L).

scholarly journals Birational classification of curves on rational surfaces

2010 ◽  
Vol 199 ◽  
pp. 43-93 ◽  
Author(s):  
Alberto Calabri ◽  
Ciro Ciliberto
Keyword(s):  
Linear System ◽  
Plane Curve ◽  
Rational Surface ◽  
Minimal Degree ◽  
Classification Theorem ◽  
Plane Curves ◽  
Cremona Transformation ◽  
Rational Surfaces

AbstractIn this paper we consider the birational classification of pairs (S, ℒ), with S a rational surface and ℒ a linear system on S. We give a classification theorem for such pairs, and we determine, for each irreducible plane curve B, its Cremona minimal models, that is, those plane curves which are equivalent to B via a Cremona transformation and have minimal degree under this condition.

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