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Components with the expected codimension in the moduli scheme of stable spin curves

Components with the expected codimension in the moduli scheme of stable spin curves 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annales UMCS, Mathematica de Gruyter

Components with the expected codimension in the moduli scheme of stable spin curves

Annales UMCS, Mathematica , Volume 69 (1) – Jun 1, 2015

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Publisher
de Gruyter
Copyright
Copyright © 2015 by the
ISSN
2083-7402
eISSN
2083-7402
DOI
10.1515/umcsmath-2015-0009
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Abstract

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.

Journal

Annales UMCS, Mathematicade Gruyter

Published: Jun 1, 2015

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