POINTED CASTELNUOVO NUMBERS 5 GAVRILFARKASANDNICOLATARASCA 1 0 2 Abstract. The classical Castelnuovo numbers count linear series of minimal degree and fixed dimension on a general curve, in the case when this number is finite. For pencils, that is, linear c seriesofdimensionone,theCastelnuovo numbersspecializetothebetter knownCatalannumbers. e UsingtheFulton-Pragaczdeterminantal formulaforflagbundles andcombinatorialmanipulations, D we obtain a compact formula for the number of linear series on a general curve having prescribed 1 ramification at an arbitrary point, in the case when the expected number of such linear series is finite. Theformulaisthenusedtosolvesomeenumerativeproblemsonmodulispacesofcurves. ] G A A linear series of type gr on a smooth curve C of genus g is a pair ℓ = (L,V) consisting of a line d . bundle L on C of degree d and a subspace of global sections V ⊂ H0(C,L) of projective dimension h r. The Brill-Noether theoremsaysthat for a generalcurveC, the varietyGr(C) of linearseriesgr on t d d a C has dimension ρ(g,r,d):=g−(r+1)(g−d+r), and is empty if ρ(g,r,d)<0. In particular,when m ρ(g,r,d)=0 there is a finite number N of linear series gr. This number is equal to g,r,d d [ r i! 2 Ng,r,d =g! . (g−d+r+i)! v i=0 Y 2 Remarkably,Castelnuovo[Cas89]correctlydeterminedN inthe1880’susingasubtledegeneration 8 g,r,d argument and Schubert calculus. However, the construction of the moduli space of curves, implicitly 8 4 assumed in the degeneration, has been achieved only in the 1960’s by Deligne and Mumford. A 0 modern rigorous proof of the Brill-Noether theorem appeared in 1980 in the work of Griffiths and 1. Harris [GH80] and is based on Castelnuovo’s original degeneration. 0 Similarly, one can consider linear series on a general curve having prescribed vanishing at a fixed 5 general point. For a smooth curve C of genus g, let p ∈ C be a point and ℓ = (L,V) ∈ Gr(C). The 1 d vanishing sequence of ℓ at p : v i (1) aℓ(p):0≤a0 <···0 admits a linear series ℓ∈Gr(C) with vanishing sequence P d aℓ(p)=a if and only if r (2) (a −i+g−d+r) ≤g. i + i=0 X Here (n) := max{n,0} for any integer n. Note that this condition is stronger than the condition + ρ(g,r,d,a)≥ 0. When (2) is satisfied, the variety of linear series ℓ ∈Gr(C) with vanishing sequence d a at the point p is pure of dimension ρ(g,r,d,a). As in the unpointed case, one can consider the zero-dimensional case. Let g,r,d be positive integers and a : 0 ≤ a < ··· < a ≤ d as above, such 0 r 2010 Mathematics Subject Classification. 14Q05(primary),14H51(secondary). Keywords and phrases. Brill-Noethertheory, enumerativegeometryonageneralcurve. 1 2 GAVRILFARKASANDNICOLATARASCA that ρ(g,r,d,a)=0. Then, by (2) the curve C admits a linear series gr with vanishing sequence a at d the point p if and only if a +g−d+r ≥ 0. When such linear series exist, their number is counted 0 by the adjusted Castelnuovo number (a −a ) i0,an(cid:1)dlets:=g−d+r. The number of linear series ℓ ∈ Gr(C) on a general curve C of genus g satisfying the condition d |ℓ(−n·p)|6=∅ at a certain unspecified point p∈C is equal to g!·n(n2−1) r i!·(n−i) n = . g,r,d,(0,1,...,r−1,n) (s−1)!(s+n−1)!(r−1)! (s−1+i)! i=2 Y Theorem 1 is proven in §1 using the determinantal formula for flag bundles. The resulting deter- minant is simplified through a series of combinatorial manipulations. As an application, we compute POINTED CASTELNUOVO NUMBERS 3 classes of closures of pointed Brill-Noether divisors in M in §2, after a result of Eisenbud and g,1 Harris. In §3 we deduce the non-proportionality of closures of Brill-Noether classes of codimension 2 in M . g WeremarkthatprovingTheorem1viaadegenerationargumentandSchubertcalculusisnotfeasi- ble. Incontrasttothesituationfrom[EH87]whereonecomputesthenumbersN byspecializing g,r,d,a to a curve having a rational component and g elliptic tails, here one would have to describe all linear series on elliptic curves having prescribed vanishing at two unspecified points (the exceptional ramification point and the point of attachment to the rest of the curve). However, unlike for 1-pointed elliptic curves, there is no adequate lower bound for Brill-Noether numbers on arbitrary 2-pointed elliptic curves. In particular we get a lot more linear series with prescribed ramification than we expect and it is difficult to determine which of these limit linear series are smoothable. 1. Counting Brill-Noether special points Let C be a general curve of genus g ≥2 and fix positive integers r and d, as well as a sequence a:0≤a <···2, the class of the divisor M (a) in M is equal to g,d g,1 r [M (a)]=µ·BN +ν·W g,d where r n 1 n µ=− g,r,d,a + n and ν = g,r,d,a . 2(g2−1) 4 g−1 g−1,r,d,(a0+1−δ0i,...,ar+1−δri) g(g2−1) 2 i=0 X Proof. Let C be a general c(cid:0)urve(cid:1)in M and consider the curve C = {[C,y]} in M obtained g y∈C g,1 by varying the point y in C. The only generator class having non-zero intersection with C is ψ, and C ·ψ = 2g−2. On the other hand, C ·Mr (a) is equal to the number of pairs (y,ℓ) ∈ C ×Gr(C) g,d d such that aℓ(y)=a, that is, n . Hence, we deduce that g,r,d,a n g,r,d,a ν = . (2g−2) g+1 2 Furthermore,let (E,p,q) be a two-pointedelliptic c(cid:0)urve(cid:1)with p−q not a torsionpoint in Pic0(E). Consider the curve D in M obtained by identifying the point q ∈ E with a moving point in a g,1 r general curve D of genus g −1. Then the intersection M (a)·D corresponds to the pairs (y,ℓ) g,d where y is a point in D and ℓ ={ℓE,ℓD} is a limit linear series with aℓE(p)= a. By [EH89, Lemma 3.4],theintersectioniseverywheretransverse. Theonlypossibilityisρ(E,p,q)=0andρ(D,y)=−1. It follows that aℓD(y) = (a0 +1−δ0i,...,ar +1−δri), for some i = 0,...,r, and in each case ℓE is uniquely determined. Studying the intersection of D with the generating classes, we obtain r g n = µ(g−1)+ν (2g−4) g−1,r,d,(a0+1−δ0i,...,ar+1−δri) 2 i=0 (cid:18) (cid:18) (cid:19)(cid:19) X whence we compute µ. (cid:3) 1 Example. When r =1, d=g−h, and a=(0,g−2h), we recover the class of the divisor M (a) g,g−h computed by Logan in [Log03, Theorem 4.5]. 3. Non-proportionality of Brill-Noether classes of codimension two In[EH87]EisenbudandHarrisshow thatallclassesof closuresofBrill-Noetherdivisorsin M are g proportional. That is, if ρ(g,r,d)=−1, then the class ofthe closure of the locus Mr of curves with g,d a linear series gr is d [Mr ]=c·BN ∈CH1(M ), g,d g where the class BN is in (9), and c is a positive rational number. If ρ(g,r,d)=−2,then the locus Mr ofcurves admitting a linear series gr is pure ofcodimension g,d d two([EH89]). Inthe caser =1,the classofthe closureofthe Hurwitz-Brill-Noetherlocus M1 has 2k,k been computed in [Tar13] using the space of admissible covers. In this section, we show that classes of Brill-Noether loci of codimension two are generally not proportionalin CH2(M ). g Thefirstnon-trivialcaseis wheng =10: inM weconsiderthe twoBrill-Noetherloci M1 and 10 10,5 M2 of codimension two. In order to show that the classes of the closures of M1 and M2 are 10,8 10,5 10,8 not proportional,we show that their restrictions to two test families are not proportional. 10 GAVRILFARKASANDNICOLATARASCA For i=2,3,let C be a generalcurve of genus i,and C a generalcurve ofgenus g−i. Consider i g−i thetwo-dimensionalfamilyS ofcurvesobtainedbyidentifying amovingpointxinC withamoving i i point y in C . The base of this family is C ×C . g−i i g−i An element C ∪ C of the family S is in the closure of M2 if and only if it admits a limit i x∼y g−i i 10,8 linear series {ℓCi,ℓCg−i} of type g28 such that ρ(i,2,8,aℓCi(x))=ρ(g−i,2,8,aℓCg−i(y))=−1. There are exactly T := n ·n i i,2,8,a g−i,2,8,(d−a2,d−a1,d−a0) a=(aX0,a1,a2) ρ(i,2,8,a)=−1 pairs (x,y) in C ×C with this property. Moreover,since the family S is in the locus of curves of i g−i i compact type, we knownthat the intersection is transverseat each point [EH87, Lemma 3.4]. Hence, we have 2 2 S · M =T =23184, S · M =T =48384. 2 10,8 2 3 10,8 3 Similarly, we compuhte i h i 1 1 S · M =2016, S · M =12096. 2 10,5 3 10,5 h 2 i 1 h i Since therestrictionof[M ]and[M ]tothe surfacesS andS arenotproportional,wededuce 10,8 10,5 2 3 2 1 that [M ] and [M ] are not proportional. 10,8 10,5 References [ACGH85] E.Arbarello,M.Cornalba,P.A.Griffiths,andJ.Harris.Geometry of algebraic curves. Vol. I,volume267 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag,NewYork,1985.4 [Cas89] GuidoCastelnuovo. Numerodelleinvoluzioni razionaligiacenti sopraunacurvadidato genere. Rendiconti R. Accad. Lincei,5(4):130–133, 1889.1 [EH86] DavidEisenbudandJoeHarris.Limitlinearseries: basictheory.Invent. Math.,85(2):337–371, 1986.2 [EH87] DavidEisenbudandJoeHarris.TheKodairadimensionofthemodulispaceofcurvesofgenus≥23.Invent. Math.,90(2):359–387, 1987.1,2,3,9,10 [EH89] David Eisenbud and Joe Harris. Irreducibility of some families of linear series with Brill-Noether number −1.Ann. Sci. Ećole Norm. Sup. (4),22(1):33–53, 1989.2,8,9 [Far13] Gavril Farkas. Brill-Noether with ramification at unassigned points. J. Pure Appl. Algebra, 217(10):1838– 1843,2013. 2 [Ful92] WilliamFulton.Flags,Schubertpolynomials,degeneracyloci,anddeterminantal formulas.Duke Math. J., 65(3):381–420, 1992.3,4 [GH80] PhillipGriffithsandJoeHarris.Onthevarietyofspeciallinearsystemsonageneralalgebraiccurve.Duke Math. J.,47(1):233–272, 1980.1 [HM82] Joe Harris and David Mumford. On the Kodaira dimension of the moduli space of curves. Invent. Math., 67(1):23–88, 1982.WithanappendixbyWilliamFulton.2 [Log03] Adam Logan. The Kodaira dimension of moduli spaces of curves with marked points. Amer. J. Math., 125(1):105–138, 2003.9 [Tar13] NicolaTarasca.Brill–Noetherlociincodimensiontwo.Compos. Math.,149(9):1535–1568, 2013.9 Humboldt-Universita¨tzu Berlin, Institut fu¨rMathematik,Unter den Linden 6,10099Berlin, Germany E-mail address: farkas@math.hu-berlin.de University of Utah,Departmentof Mathematics,155S1400E, SaltLake City,UT 84112,USA E-mail address: tarasca@math.utah.edu

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