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Here we study the BrillNoether theory of "extremal" Cornalba's theta-characteristics on stable curves C of genus g, where "extremal" means that they are line bundles on a quasi-stable model of C with (Sing(C)) exceptional components. 1. Introduction. For any integer g 2 let M g denote the moduli space of stable curves of genus g over an algebraically closed field K such that char(K) = 0. Fix any Y M g . The topological type (if K = C) or the equisingular type (for arbitrary K) may be described in the following way. Fix an ordering Y1 , . . . , Ys of the irreducible components of Y . The type is uniquely determined by the string of integers listing the geometric genera of Y1 , . . . , Ys , the integers (Sing(Yi )), 1 i s, and the integers (Yi Yj ), 1 i < j s (see [1], p. 99). Recently, the BrillNoether theory of thetacharacteristics of smooth curves had a big advances due to a solution by L. Benzo ([3]) of a conjecture of G. Farkas ([6], Conjecture 3.4). In this note we show that such a result may be used for the study of the BrillNoether theory of Cornalba's theta-characteristics on M g . Indeed, we will check that for the extremal theta-characteristics we are looking for in this note the existence of such a theta-characteristic on Y with prescribed number of 2010 Mathematics Subject Classification. 14H10; 14H51; 14H42. Key words and phrases. Stable curve, theta-characteristic, spin curve, BrillNoether theory. The author was partially supported by MIUR and GNSAGA of INdAM (Italy). E. Ballico linearly independent sections, r + 1, is equivalent to the existence of thetacharacteristics E1 , . . . , Es on the normalizations C1 , . . . , Cs of Y1 , . . . , Ys and with s h0 (Ci , Ai ) = r + 1. i=1 Let Sg , g 2, be the set of all theta-characteristics on smooth genus g curves, i.e. the set of all pairs (C, L) with C Mg , L Pic(C) and r L2 C . For all integers r -1 set Sg := {(C, L) Sg : h0 (L) = = r is a locally closed subset of S and each point of it has r + 1}. The set Sg g codimension at most r+1 in Sg ([8], part (ii) of Theorem 1.10). Maurizio 2 Cornalba proved the existence of a compactification S g of Sg equipped with a finite morphism ug : S g M g such that each fiber of ug has cardinality 22g ([5], Proposition 5.2 and first part of §3). There are many topological types for which the BrillNoether theory of theta-characteristics with r + 1 linearly independent sections never occurs in the expected codimension, i.e. in codimension r+1 (see [2] for a description of all theta-characteristics 2 with g linearly independent sections). The claim of this note is that to study the BrillNoether theorem of S g \ Sg one needs to distinguish the quasi-stable model on which a Cornalba's theta-characteristic lives as a line bundle. In other compactifications of Sg (as in [9]) torsion-free sheaves are used; prescribing the non-locally free points of these sheaves on some C M g is equivalent to prescribe the images in Sing(C) of the quasistable model of C on which a Cornalba's theta-characteristic "is" a line bundle (it is not quite a line bundle L, but a line bundle up-to inessential isomorphisms and we also need to prescribe the line bundle L2 ([5], Lemma 2.1 and first part of §3)). None of these problems affect the BrillNoether theory for the theta-characteristics we will consider in this note (we call them the maximally singular ones). For these theta-characteristics the computation of h0 is reduced to the computations of h0 for theta-characteristics on the normalizations of all the irreducible components of the given C M g . Hence the existence part is reduced to an existence part on smooth curves for all genera up to g. There is a natural injective morphism from S g into Caporaso's compactification P g-1,g ([4]) of the set of all degree g - 1 line bundles on Mg ([7]). A Cornalba's theta-characteristic associated to a stable curve C is said to be maximally singular if it is a line bundle on the quasi-stable model C of C obtained blowing up all singular points of C. A Cornalba's theta-characteristic on C is maximally singular if and only if it induces a theta-characteristic on the normalization of C ([5], Lemma 1.1). If C has compact type, then each theta-characteristic on C is maximally singular, because for each S Sing(C), the quasi-projective curve C \ S has (S) + 1 connected components. Obviously a = 0 for a = 0, 1. Define the function : N N in the 2 following way. Set (0) := 1 and (1) := 1. For all integers q 2 let (q) Components with the expected codimension... be the maximal positive integer such that and (3) = 2. (q)+1 2 q. We have (2) = 1 Theorem 1. Fix a type for genus g stable curves. Let q1 , . . . , qs be the geometric genera of the irreducible components of stable curves with type . Fix integers ai , 1 i s, such that 0 ai (qi ) for all i and set r := -1 + s ai . Then there is an irreducible component of the set of i=1 all maximally singular Cornalba's theta-characteristics for stable curves with ai type with codimension s i=1 2 and such that for a general (Y, L) with Y = Y1 · · · Ys , each Yi of geometric genus qi and h0 (Ci , L|Ci ) = ai for all i, where Ci is the normalization of Yi . In most cases no component satisfying the thesis of Theorem 1 may be smoothable, i.e., it is in the closure inside S g of an irreducible component r of Sg , just because r may be very high. 2. The proof. Remark 1. Fix an integer q 0 and a smooth genus q curve D. If q 3, then assume that D is general in its moduli space. A corollary of Gieseker Petri theorem (case q 3) ([1], Proposition 21.6.7) or RiemannRoch gives that every theta-characteristic A on D satisfies h0 (D, A) 1. We will only use the existence of theta-characteristics A, B on D such that h0 (D, A) = 0 and h0 (D, B) = 1. 1 Remark 2. Notice that S3 has codimension 1 in M3 , because the hyperel1 liptic locus of M3 has dimension 5. By [6], Theorem 1.2, Sg has a component of the expected codimension, 1, for all g 3. Lemma 1. Let Y be a reduced projective curve such that Y = CT such that T P1 , (C T ) = 2 and each point of C T is a nodal point of Y . Let R be = any line bundle on Y such that deg(R|T ) = 1. Then hi (Y, R) = hi (C, R|C), i = 0, 1. Proof. We have the MayerVietoris exact sequence: (1) 0 R R|C R|T R|C T 0 Since deg(C T ) = 2, deg(R|T ) = 1 and R is a line bundle, the restriction map H 0 (T, R|T ) H 0 (C T, R|C T ) is an isomorphism. Hence (1) gives hi (Y, R) = hi (C, R|C), i = 0, 1. Proof of Theorem 1. Fix a stable curve Y = Y1 · · · Ys with each Yi of geometric genus qi . Let C = C1 · · · Cs be the normalization of Y with Ci the normalization of Yi . Assume for the moment the existence of a theta-characteristic Ai on Ci such that h0 (Ci , Ai ) = ai and let A be the line bundle on C1 · · · Cs with A |Ci = Ai for all i. Let Y be the quasistable curve with Y as its stable reduction and with (Sing(Y )) exceptional components. Let A be any line bundle on Y with A as its pull-back to E. Ballico C and deg(A|J) = 1 for each exceptional component J of Y . Applying (Sing(Y )) times Lemma 1, we get h0 (Y , A) = r + 1. A is a totally singular Cornalba's theta-characteristic. Now we count the parameters. By the +1 definitions of the integers (qi ) and ai we have qi ai2 for all i if ai 2. a By [3], Theorem 1.2, there is an irreducible component i Sqii -1 if ai 2. For the case ai = 0 use Remark 1. For the case ai = 1 use Remark 2. Taking all (Y, A) coming from all (Ci , Ai ) i , we get a family of curves Y ai with codimension s i=1 2 in the subset M ( ) M g with type . This is r a maximal family (i.e. an open subset of an irreducible component of S g ), because each i is a maximal family and for all Y M ( ) the fiber u-1 (Y ) g has the same number of elements.
Annales UMCS, Mathematica – de Gruyter
Published: Jun 1, 2015
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