The genus of an algebraic curve X is equal, by definition, to the genus of the complete algebraic curve birationally isomorphic to X. For any integer g > 0 there exists an algebraic curve of genus g The genus of a plane curve 1 A formula for the genus of a nice plane curve The genus g of a nonsingular plane curve of degree d equals d−1 2. The genus g of a plane curve of degree d with only ordinary multiple points equals g = d−1 2 − X P m(P) 2 where the sum is over the multiple points P (with multiplicity m(P))

- In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus g > 1, given by an equation of the form y 2 + h (x) y = f (x
- Genus of a Curve. a number characterizing an algebraic curve. The genus of the nth degree curve f (x, y )= 0 is. where r is the number of double points. When more complex singular points are present, they are counted as the corresponding number of double points; for example, a cusp is counted as one double point, and a triple point is counted as.
- It has a singular point at the origin of order . Now, a non-singular curve of degree in has genus (see elliptic curve ), but this curve doesn't: in fact, using the birational map we see that this curve is birational to , hence has genus
- There are two related definitions of genus of any projective algebraic scheme X: the arithmetic genus and the geometric genus. When X is an algebraic curve with field of definition the complex numbers , and if X has no singular points , then these definitions agree and coincide with the topological definition applied to the Riemann surface of X (its manifold of complex points)
- In classical algebraic geometry, the genus-degree formula relates the degree d of an irreducible plane curve with its arithmetic genus g via the formula: = ()
- Recall the genus formula g = ( d − 1 2) − ∑ m p ∈ S ( m p 2) where S is the set of singular points on the curve, and m p is the multiplicity of point p. There is a catch of sorts: the multiplicity is not in general the same as what one obtains from solving the appropriate polynomial system to find the singularities
- ag.algebraic geometry - Computing the genus of a plane curve - MathOverflow 0 Let b (x) = x 4 + 3 x 3 + 3 x 2 + 2 x + 1, and let a (x) ∈ Z [ x] be a separable polynomial. Let C be the plane curve defined by (y 2 + (x + x 2 + x 3) a (x)) 2 − a (x) 2 b (x) = 0

- The genus of a curve is a birational invariant which plays an important role in the parametrization of algebraic curves (and in the geometry of algebraic curves in general). In fact, only curves of genus 0 can be rationally parametrized. So in the process of parametrization we will first compute the genus of the curve C
- Curve Genus. One of the Plücker characteristics , defined by. where is the class, the order, the number of nodes, the number of cusps, the number of stationary tangents ( inflection points ), and the number of bitangents . SEE ALSO: Riemann Curve Theorem REFERENCES
- 53.8 The genus of a curve can be negative, and depends on the base field and should be written
- EXAMPLES OF GENUS 5 CURVES 1. Genus 5 curves in P2 Example 1.1. A degree 5 plane curve with one node. Indeed, by the degree-genus formula, p g = (5 1)(5 2) 2 1 = 5 A speci c example of such a curve is V(x3yz+ y5 + z5), with node at the point [1 : 0 : 0]. Proposition 1.2. A curve Xof genus 5 can be represented as a plane quintic with one node i it has a g1 3
- can occur such that a nonsingular algebraic curve C ⊆ P3 with degree dand genus gexists. Classiﬁcation for curves with low genus, namely 0,1, and 2, are rather easy to deal with, and we will do this. For curves with higher genus, a fruitful approach has been gained by asking the least degree of a surface in P3 on which curve lies
- De nition 1.1. The projective curve Ein P2 is called an elliptic curve. Since the projective curve Eis de ned by a homogeneous polynomial of degree 3;by genus degree formula, the genus of Eis g= (3 1)(3 2)=2 = 1: Simiarly, every plane curve can be embedded into a projective curve. Suppose Cis a plane curve de ned by f(x;y) = 0 with degf(x;y.

Figure 3: The complex points a curve Cof genus g From an algebraic geometry point of view, the genus of a curve C2Mcan be de ned as the dimension4 g(C) = dimH0(C; 1 C) (4) of the space of sections of the cotangent line bundl Curves of maximal genus 68 Lecture 15. October 21, 2011 70 15.1. Beyond Castelnuovo 70 15.2. Analogues of Castelnuovo 71 Lecture 16. October 26, 2011 73 16.1. In ectionary points 73 16.2. Pluc ker formula 75 Lecture 17. November 2, 2011 78 17.1. Osculating linear spaces 78 17.2. Plane curves 79 Lecture 18. November 4, 2011 83 18.1. A remark about in ectionary points 83 18.2. Weierstrass points. (1)degree 1 genus 1 curve is contracted to a point, (2)degree 2 genus 1 curve is mapped 2 to 1 onto P1, (3)degree 3 genus 1 curve is mapped into P2as a plane cubic, (4)degree 4 genus 1 curve is the intersection of two quadrics in P3, () e genus of a smooth curve over C determines its topology. A smooth curve over C leads a double life,rst as an algebraic variety of dimension, second as a complex manifold of dimension. e complex manifold corresponding to the anelineA overCisthecomplexplaneC,topologicallyequaltoR. eprojective line P = A ∪ {∞} is the one-point compactication C ∪. The genus g is determined by the condition that the dimension of Θ(x ) is n 1ν −g +1, as is made plausible by the following considerations. Each value of x corresponds to n 1 values of y (the roots of χ(α,y) where α is the given value for x), which is to say that each value of x occurs at n 1 points of the curve. In particular, x has

By definition, the genus of an algebraic curve is equal to the genus of its non-singular model. For any non-negative integer $ g $ there exists an algebraic curve of genus $ g $ . Rational curves are distinguished by the equality $ g = 0 $ . If $ X $ is a projective plane curve of order $ m $ , then $$ \pi = \frac{( m - 1 ) ( m - 2 )}{2. The curves of genus ≥2 are much more difﬁcult to work with, and the theory is much less complete. One result that illustrates the difference between this case and the genus 1 case is Faltings' theorem, which states that for curves deﬁned over Q, the set of rational points is ﬁnite; but no practical algorithm is yet known for ﬁnding them. 1.1 Deﬁnitions: Elliptic curves and the. * Genus of a curve: lt;div class=hatnote|>For the term in index theory, see |Genus of a multiplicative sequence|*.| World Heritage Encyclopedia, the aggregation of. **curve** **of** drying [TECH.] die Trocknungskurve **curve** **of** sliding [TECH.] die Gleitkurve Pl.: die Gleitkurven course of the **curve** der Kurvenverlauf Pl.: die Kurvenverläufe flattening of the **curve** Verflachung der Kurve run of the **curve** Verlauf der Kurve radius of the **curve** [BAU.] der Krümmungsradius [Schalungsbau] degree of **curve** [GEOL.] der Kurvengrad Pl.: die Kurvengrad

- Given an elliptic genus with non-degenerate parameters. δ, ϵ ∈ ℂ. \delta, \epsilon \in \mathbb {C} (as above, see also at j-invariant ), the Jacobi quartic Riemann surface which is given by the equation. Y 2 = X 4 − 2 δ X 2 + ϵ. Y^2 = X^4 - 2 \delta X^2 + \epsilon. is naturally parameterized by the upper half plane
- The genus of a connected surface (i.e. a connected topological space any point of which has a neighborhood homeomorphic to the plane) is the maximum number of simple closed curves without common points that can be traced inside this surface without rendering the resultant manifold disconnected (i.e. such that the complement of these curves remains connected); in concrete terms, if we consider.
- THE GENUS OF CURVE, PANTS AND FLIP GRAPHS 3 Once we quotient the graphs by their automorphisms, the resulting graphs are nite so they have nite genus. Let g be a closed genus gsurface and g;1 be a surface of genus gwith a single marked point. As mentioned above, the theorem of Ringel and Youngs about the genus of complete graphs and some simple observations about topological types of curves.
- Given a plane curve we can compute the genus from the number ofnodes: Theorem 23Given an irreducible curveC=V(f),f∈k[x, y],d= degfwithδordinary double points, the genusg(C)is (d1) (d2
- (Geometric) Genus of a curve equals to -1 in SageMath. edit. genus. asked 2019-12-08 12:57:00 +0200. azerbajdzan 102 2 7 11. updated 2019-12-08 13:07:09 +0200. What does it mean when Sage computes genus of a projective plane curve to be -1? Input: x,y,z = QQ['x,y,z'].gens() C = Curve(x^4 + 10*x^2*y*z + 5*y^2*z^2) C.genus() Output:-1 edit retag flag offensive close merge delete. Comments. Not a.
- De nition 1.1. The projective
**curve**Ein P2 is called an elliptic**curve**. Since the projective**curve**Eis de ned by a homogeneous polynomial of degree 3;by**genus**degree formula, the**genus****of**Eis g= (3 1)(3 2)=2 = 1: Simiarly, every plane**curve**can be embedded into a projective**curve**. Suppose Cis a plane**curve**de ned by f(x;y) = 0 with degf(x;y.

Computes the geometric genus of a plane curve. If ib is specified (and only then) we assume that I has the following properties: Denote the variables of R=ring(I) by v,u,z. All singularities of C have to lie in the chart z!=0 and the curve should not contain the point (1:0:0). Furthermore we assume that ib has the following properties: The entries are in K(u)[v] inside frac(R) where the i-th. Welcome to the LMFDB, the database of L-functions, modular forms, and related objects. These pages are intended to be a modern handbook including tables, formulas, links, and references for L-functions and their underlying objects

Every genus 1 curve can be written as a cubic plane curve. Thus we've \seen all genus 1 curves. By special request, we proved that: Proposition. There is a natural bijection between the degree 1 line bundles on a genus 1 curve Cand the points of the curve. Proof. The link is as follows. If pis a point on the curve, then O C(p)isa degree 1 invertible sheaf. If Lis a degree 1 invertible sheaf. Idea. Under the function field analogy it makes sense to regard a number field as the rational functions on an arithmetic curve over F1. Accordingly, there is a sensible version of the concept of genus of a curve for number fields.. This genus of a number field was originally introduced in in a maybe somewhat ad hoc way.It is derived as being that definition which makes the Riemann-Roch. Curves of genus 5 23 12. 9/28/15 24 12.1. Canonical curves of genus 5 24 12.2. Adjoint linear series 25 13. 9/30/15 27 13.1. Program for the remainder of the semester 27 13.2. Adjoint linear series 27 14. 10/2/15 29 15. 10/5/15 32 15.1. Castelnuovo's Theorem 33 16. 10/7/15 34 17. 10/9/15 36 18. 10/14/15 38 1. 2 AARON LANDESMAN 19. 10/16/15 41 19.1. Review 41 19.2. Minimal Varieties 42 20. 10.

- ed by the degree, d, by the formula g= (d 1)(d 2) 2 [7]. Therefore curves in three-dimensional space are more interesting. It turns out that even though the genus is not completely deter
- 53.10 Curves of genus zero. Later we will need to know what a proper genus zero curve looks like. It turns out that a Gorenstein proper genus zero curve is a plane curve of degree $2$, i.e., a conic, see Lemma 53.10.3.A general proper genus zero curve is obtained from a nonsingular one (over a bigger field) by a pushout procedure, see Lemma 53.10.5..
- A curve of genus 3 is hyperelliptic or else it is isomorphic to a plane curve of degree 4. genus 4. When a non-hyperelliptic curve Y has genus g = 4, the canonical map embeds it as a curve of degree 2g − 2 = 6 into P3. Let's call the image Y too. We try to ﬁnd the degrees of the deﬁning equations for Y. Let's choose for K an eﬀective divisor. Then the global sections of O(K) are.
- All genus 2 curves are hyperelliptic, meaning any one can be de ned by giving 6 points in P1. The curve is then the double cover of P1 branched at the 6 points. If the curve is given in the form y2 = f(x) For a polynomial f of degree 5 or 6, then the points are precisely the roots of f, plus possibly the point at in nity depending upon if f is degree 5. The space Mtr 0;6 is one we know well.
- It was first pointed out by Weil that we can use classical invariant theory to compute the Jacobian of a genus one curve. The invariants required for curves of degree n = 2,3,4 were already known to the nineteenth centuary invariant theorists. We have succeeded in extending these methods to curves of degree n = 5, where although the invariants are too large to write down as explicit.
- We analyze GIT stability of nets of quadrics in $\\mathbb{P}^4$ up to projective equivalence. Since a general net of quadrics defines a canonically embedded smooth curve of genus five, the resulting GIT quotient gives a birational model of the moduli space of genus 5 curves. We study the geometry of the associated contraction and prove that the constructed GIT quotient is the final step of the.
- for elliptic curves, the CRT method for genus 2 curves is one of several known methods for constructing curves and we will brie y describe some of the other approaches as well. 2. Generating elliptic curves with a prescribed number of points In this section we give an algorithm for the following problem (under certain conditions): Given a prime number 'and a positive integer N, construct an.

- t |D| ≤ 10,000,000. The computation of the hyperelliptic curves was achieved using the methods described in the paper A database of genus 2 curves over the rational numbers which are applicable to hyperelliptic.
- Curves of Genus 2 with Infinite Pro-Galois Covers 3 the knowledge of the authors, no examples of non-constant curves of genus 2 with a pro-Galois curve cover have been known so far. 1.2 The Main Result In this subsection we shall formulate the main result of this paper. We thereby freely use some results of [1] which we recall at the beginning of Section 3. We begin with some notation. Let pbe.
- curves of genus g≥ 3 1 and outline the construction of the (coarse) moduli scheme of stable curves due to Gieseker. The notes are broken into 4 parts. In Section 1 we discuss the general problem of constructing a moduli space of curves. We will also state results about its properties, some of which will be discussed in the sequel. We begin Section 2 by recalling from [DM] (see also [Vi.

genus curves \in nature: as covers of other curves, as divisors on surfaces, and so on. Since we care about arithmetic, in this section we do not assume that the ground eld is algebraically closed, and pay careful attention to elds of de nition. Chapter II: Arithmetic. The main theme of this section is techniques for understanding the rational points on curves de ned over number elds. Canonical curves of genus eight Shigeru MUKAI and Manabu IDE y Let Cbe a smooth complete algebraic curve of genus gand C 2g 2 ˆPg 1 the canonical model. It is generally di cult to describe its equations for higher genus. We restrict ourselves to the case of genus 8. If Chas no g2 7, then C 14 ˆP7 is a transversal linear section [G(2;6) ˆP14] \H 1 \\ H 7 of the 8-dimensional Grassmannian.

smooth curve of (arithmetic) genus g 2 and a theta characteristic Lon C, i.e. a line bundle Lon Csuch that L 2 is isomorphic to the canonical bundle ! C. This compacti cation is compatible with the Deligne-Mumford compacti cation M gof the coarse moduli space M gof smooth curves of genus gvia stable curves [DM69]. In particular there exists a natural morphism ˇ: S g!M gwhich sends the mod-uli. of smooth genus 3 curves by smooth degree 5 curves. As we said, for the smooth generic ber, Bruin [8] gives explicitly the hyperelliptic curve Xwhose Jacobian is the Prym of the cover. Using a particular deformation, we show that we can specialize the equation of Xto nd X. Acknowledgements We want to thank Jeroen Sijsling for his comments on an earlier version of the paper and Abhinav Kumar. of curves over ﬁnite ﬁelds, with particular emphasis on curves of small genus. We prove that for every ﬁxed g≥2, the discrete logarithm problem in degree 0 class groups of curves of genus gcan be solved in an expected time of O˜(q2−2g), where F q is the ground ﬁeld. This result generalizes a corresponding result for hyperelliptic curves given in imaginary quadratic representation. Elliptic genera as super p p-brane partition functions. The interpretation of elliptic genera (especially the Witten genus) as the partition function of a 2d superconformal field theory (or Landau-Ginzburg model) - and especially of the heterotic string (H-string) or type II superstring worldsheet theory - originates with:. Edward Witten, Elliptic genera and quantum field theory.

genus one curves, due to Selmer. Let Cbe a plane curve de ned by the cubic 3x3 + 4y3 + 5z3 = 0 The curve C is smooth and of genus one, admits points over every completion of Q, but no Q-points. However it clearly admits a point over a small cubic extension of Q. orF example, we see that it has a point over Q(3 The significance of this genus value lies in the fact that only curves of genus 0 can be expressed parametrically using rational polynomials. Genus and degree equations are also derived for intersection curves involving surface patches with simple base points. A class of surfaces is identified for which any plane section is a rational curve. AB - It is shown that two generic triangular surface.

Curves of genus two with many automorphisms (general theory) In this section, we again assume char. k = p ~ 3. We recall results by Igusa [9]. Every curve C defined by (1.1) is in a unique way a two-sheeted. 130 covering of the projective line pl. We denote by i the automorphism of C which. LOG SURFACES, COVERS, AND CURVES OF GENUS 4 3 Theorem 2. The forgetful map : X 0!M 4 induces an isomorphism X 0 ˘=Bl HM 4: Thus, X 0 provides a modular interpretation of the blowup of the hyperelliptic locus in M 4.The map : X 0!M 4 does not extend to a regular map from X 0 to any known modular compacti cation of Split jacobian curves of genus 2 have also been used to exhibit nonisomorphic curves with the same jacobian; vide [3, 4]. The approach in these papers is through the algebraic geometry of abelian varieties, and the constructions are therefore far from explicit. Consider the following: Let X be a curve of genus 2, and f:X—>Ea map fro

** Previous results on genera g of F_{q^2}-maximal curves are improved: (1) Either g\leq (q^2-q+4)/6, or g=\lfloor(q-1)^2/4\rfloor, or g=q(q-1)/2; (2) The hypothesis on the existence of a particular Weierstrass point in \cite{at} is proved; (3) For q\equiv 1\pmod{3}, q\ge 13, no F_{q^2}-maximal curve of genus (q-1)(q-2)/3 exists; (4) For q\equiv 2\pmod{3}, q\ge 11, the non-singular F_{q^2}-model**. is a curve of genus 1, we get a Prym variety Pr(Z=X) of dimension 2 that is isogenous to the Jacobian of a curve Yof genus 2. In [26] Ritzenthaler and Romagny gave an explicit equation of the curve Yin terms of the equations for the curves Xand Zin the case that Zis a non-hyperelliptic curve. Goals and results The main goal of this Ph.D.-thesis is to reverse the construction by Ritzenthaler. Our technique is inspired by the geometric description of the group law and applies to hyperelliptic curves of arbitrary genus. While Cantor's general composition involves arithmetic in the polynomial ring \(\mathbb{F}_q[x]\) , the algorithm we propose solves a linear system over the base field which can be written down directly from the Mumford coordinates of the group elements Hyperelliptic curves with small embedding degree and large prime-order subgroup are key ingredients for implementing pairing-based cryptographic systems. Using our closed formulas for the Jacobian order, we propose two algorithms which complement those of Freeman and Satoh to produce genus 2 pairing-friendly hyperelliptic curves. Our method. [Ca] G. Castelnuovo, Ricerche di geometria sulle curve algebriche, Zanichelli, Bologna, 1937. [C] C. Ciliberto, Hilbert functions of finite sets of points and the genus of a curve in a projective space, Space curves (Rocca di Papa, 1985), Lecture Notes in Math., vol. 1266, Springer, Berlin, 1987, pp. 24-73

OF THE SPACE OF CURVES OF GIVEN GENUS by P. DELIGNE and D. MUMFORD (1) Fix an algebraically closed field k. Let Mg be the moduli space of curves of genus g over k. The main result of this note is that Mg is irreducible for every k. Of course, whether or not M s is irreducible depends only on the characteristic of k. When the characteristic is o, we can assume that k ~- (1, and then the result.

of a curve of genus 2 whose jacobian variety has complex multiplication. Example 1.2. Let us exhibit an inﬁnite family of genus 2 curves with CM such that the endomorphism ring is the ring of algebraic integers in a cyclic extension of Q that contains Q(p 5). Suppose p ⌘ 1 mod 12 is a prime, then f = x4 +10px2 +5p2 has roots (1.1) ± q p(5±2 p 5). So the splitting ﬁeld K = Kp of f over. Syzygies of Prym and paracanonical curves of genus 8 Elisabetta Colombo, Gavril Farkas, Alessandro Verra, and Claire Voisin Abstract. By analogy with Green's Conjecture on syzygies of canonical curves, the Prym-Green conjecture predicts that the resolution of a general level pparacanonical curve of genus gis natural. The Prym-Green Conjecture is known to hold in odd genus for almost all.

- e the corresponding newforms and present.
- SHIMURA CURVES OF GENUS AT MOST TWO JOHN VOIGHT Abstract. We enumerate all Shimura curves XD 0 (N) of genus at most two: there are exactly 858 such curves, up to equivalence. The elliptic modular curve X 0(N) is the quotient of the completed upper half-plane H by the congruence subgroup 0(N) of matrices in SL 2(Z) that are upper triangular modulo N 2Z >0. The curve X 0(N) forms a coarse moduli.
- Let ^2. A stable curve of genus g over S is a proper flat morphism n : C->S whose geometric fibres are reduced, connected, i-dimensional schemes Gg such that: (i) Gg has only ordinary double points; (ii) ifE is a non-singular rational component ofCg, then E meets the other components of C in more than 2 points; (iii) dimH^J^. We will study in this section three aspects of the theory of stable.
- consider curves of genus one in certain multiprojective spaces and provide explicit equations for their Jacobians, provided that their Jacobians have certain level structure. I do this in the three cases of P2; P1 £ P1, and P4: This problem has been and is currently being worked on by a number of people. This includes Salmon, who in 1873 published a book which contains formulas that.

ON THE CONVEX HULL GENUS OF SPACE CURVES-t J. H. HUBBARD (Received for publication 5November 1979) LET K C R3 be a simple closed curve, and K its convex hull. In [l], Almgren Thurston define the (oriented) convex hull genus of K to be the minimal genus of an (oriented) surface contained in g and bounded by K. They give examples showing that even if K is unknotted both the orientable and non. Complex curves are classified by an invariant called the genus, which measures the number of holes. The skin of a bagel is an example of a genus one curve. I'm going to introduce families of curves of genus g and, by giving lots of examples, motivate how one can learn about a particular curve by viewing it as a member of a family

\end{enumerate} The above results provide some new evidences on maximal curves in connection with Castelnuovo's bound and Halphen's theorem, especially with extremal curves; see for instance the conjecture stated in Introduction A genus one curve deﬁned over Qwhich has points over p for all primes p may not have a rational point. It is natural to study the classes of Q-extensions over which all such curves obtain a global point. In this article, we show that every such genus one curve with semistable Jacobian has a point deﬁned over a solvable extension of Q.

In this paper we begin to study curves on a weighted projective plane with one trivial weight, P(1, m, n), by determining the genus of curves of Fermat type. These are curves, C, defined by the homogeneous polynomial xamn 0 + xan 1 = xam 2. We begin by finding local coordinates for the standard affine cover of P(1, m, n), and then prove that the curve is smooth. This is done by pulling. Academia.edu is a platform for academics to share research papers On a Curve of Genus 7 @article{Macbeath1965OnAC, title={On a Curve of Genus 7}, author={A. Macbeath}, journal={Proceedings of The London Mathematical Society}, year={1965}, pages={527-542} } A. Macbeath; Published 1965; Mathematics; Proceedings of The London Mathematical Society; View via Publisher. Save to Library. Create Alert . Cite. Launch Research Feed. Share This Paper. 76 Citations. A GENUS ONE CURVE OF ARBITRARY DEGREE TOM FISHER ABSTRACT. We extend the formulae of classical invariant theory for the Jacobian of a genus one curve of degree n 4 to curves of arbitrary degree. To do this, we associate to each genus one normal curve of degree n, an n n alternating matrix of quadratic forms in n variables, that represents the invariant differential. We then exhibit the. CURVES OF GENUS 2 WITH SPLIT JACOBIAN ROBERT M. KUHN ABSTRACT. We say that an algebraic curve has split jacobian if its jacobian is isogenous to a product of elliptic curves. If X is a curve of genus 2, and f: X t E a map from X to an elliptic curve, then X has split jacobian. It is not true that a complement to E in the jacobian of X is uniquely determined, but, under certain conditions.

Theorem 0.4 of Matrix Factorizations and families of curves of genus 15 descibes a dominant construction of curves of genus 15. The only non-unirational step is the choice of 6 points on a auxiliar curve E of genus 11, which actually is a Chang-Ran curve of degree 12 in P 3.We choose these points by decomposing a random hyperplane section into its irreducible components over kk CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): Abstract. In this paper we begin to study curves on a weighted projective plane with one trivial weight, P(1, m, n), by determining the genus of curves of Fermat type. These are curves, C, defined by the homogeneous polynomial xamn 0 + xan 1 = xam 2. We begin by finding local coordinates for the standard affine. curves of genus gwith npunctures (see a de nition below) is not only an operad, but a modular operad. In fact, the moduli spaces M g;n became a model example for Getzler and Kapranov [?] to de ne the notion of a modular operad. We will mostly concentrate on the genus zero operad M 0;n and, moreover, on a real version of it. The moduli space M 0;n of compact complex algebraic curves of genus. To get more evidences for general conjectures raised in [11], which are on ∆-genus inequality and on the integrality of pregemetric shells (cf. Conjecture 1.1), we investigate the pregeometric shells of a canonical curve of genus ≤ 5 Generalized explicit descent and its application to curves of genus 3 3 expected for a general curve. For a general genus 3 curve over Q, our method requires the class group of a number ﬁeld of degree 28; this seems to be the smallest possible, given that 28 is the smallest index of a proper subgroup of Sp6.F2/. For general genus 4 curves.

Every curve C/kof genus 2 is hyperelliptic over k. In general, we will implicitly assume that curves of genus 2 over k are given by hyperelliptic equations. The iso-morphisms between two such curves correspond, in terms ofhyperelliptic equations, to transformations of the type (1.2) X = aX +b cX +d,Y = (ad−bc)Y (cX +d)3, associated to a. Genus Zero and One For curves Cofgenus 0, the problem is completely solved: Wecan decideif Chas rational points or not, and if so, theycan be parametrizedby a map P1!C. If Chasgenus 1, the situation is less favorable. Methods are not known to always work (but do so in principle modulo standard conjectures

Aim: Translate the existence of genus 2 curves into a problem about quadratic forms. Let A be an abelian surface (dim(A) = 2), NS(A) = Div(A)/≡ its N´eron-Severi group. Observation: If C ⊂ A is a (smooth) curve of genus 2, then C2 = 2 and so its class θ C = cl(C) ∈ NS(A) is a principal polarization on A. The converse is false: not every θ ∈ P(A) := {principal po-larizations on A. A general curve Cof genus 7 can be realized in P2 as a curve of degree 7 with eight nodes. The blow-up of P2 at the nodes of C embeds into P6 by the linear system of quartic curves vanishing at the nodes of C. The surface has degree 8 and contains C can [ACGH]. We, therefore, conclude that the minimal degree surface containing a canonical curve of genus 7 can have degree 5;6;7 or 8. The. perelliptic curves of genus gde ned over a eld of characteristic pwith a-number equal to g 1. These results show that for a hyperelliptic curve with a= g 1, the bound on the genus is even lower than was previously known. We must actually have g<pfor such a curve to exist. Based on computations for p= 5 and p= 7, it seems possible that this bound may be even lower when p>3. 2. When g= p 1, for.

Moduli of genus two curves. It follows that genus two curves can be classified, or at least parametrized. That is, an isomorphism class of a genus two curve is precisely given by six distinct (unordered) points on , modulo automorphisms of . In other words, one takes an open subset , and quotients by the action of . In fact, this is a description of the coarse moduli space of genus two curves. Call a curve X tamely superelliptic if X is a cyclic cover of P1 of degree not divisible by p, and g(X) ≥ 2. These are the curves of genus ≥ 2 with equations of the form yn = f(x) with p - n. Corollary 1.9. If X is tamely superelliptic, then X ⇒ H for some hyperelliptic curve H. Proof. Theorem 1.7 applies because the Galois group is. Although the zero set of this polynomial is generically a curve of genus 2, it seems that m(P k) is a rational multiple of L0(E,0) for k∈ Z ≤4 −{−1} with Ea certain elliptic curve that is a factor of the jacobian of the genus 2 curve. Also it appears that m(P k) is a rational multiple of L0(χ,−1) for k∈ {−1,8} with χa certain real-valued Dirichlet character. In this thesis we

A Teichmu¨ller curve is primitive if it does not arise from a curve in Mh, h < g, via a covering construction. Our main result completes the classiﬁcation of primitive Teichmu¨ller curves in genus two. Theorem 1.1 The decagon form ω = dx/y on the curve y2 = x(x5 −1) gen-erates the only primitive Teichmu¨ller curve f : V →M2 coming. Although I am aware that the genus has these two different definitions I have not yet studied the cases where they will differ since they coincide, I believe, when the curve is nonsingular. I would appreciate it very much if you could elaborate. In particular, does the Riemann-Hurwitz formula refer to the arithmetic or geometric genus purely trigonal curves of genus four in [6], a paper which draws heavily on the re-sults presented here. It is perhaps useful to compare and contrast these two cases. As demonstrated in Schilling's generalization of the Neumann system [28],there are basi-cally two cases of trigonal cyclic covers,the order of a related linear diﬀerential oper- ator that commutes with the given one of order. Curves of genus four attain a performance level comparable to elliptic curves. A large choice of curves is therefore available for the deployment of curve-based cryptography, with curves of genus three and four providing their own advantages as larger cofactors can be allowed for the group order