POINTED CASTELNUOVO NUMBERS 5 GAVRILFARKASANDNICOLATARASCA 1 0 2 Abstract. The classical Castelnuovo numbers count linear series of minimal degree and fixed di- mension 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 <···<ar ≤d X is the ordered sequence of distinct vanishing orders of sections in V at the point p. Given r,d and r a a sequence a : 0 ≤ a < ··· < a ≤ d as in (1), the adjusted Brill-Noether number is defined as 0 r ρ(g,r,d,a):=ρ(g,r,d)− r (a −i). Eisenbud and Harris ([EH87, Proposition 1.2]) proved that a i=0 i general pointed curve (C,p) of genus g >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 ) i<j j i (3) N =g! . g,r,d,a r (g−d+r+a )! i=Q0 i Inorderto prove(3), onecanspecializethe generalcurveofgenusg toarationalcurvewithg elliptic Q tails attached to it, specialize the marked point to a point on the rational component, and count via Schubert calculus degenerations of linear series on this singular curve (see the proof of Proposition 1.2 in [EH87]). From (2), it follows that if a is the vanishing sequence at a general point of a linear series gr on d the general curve, then necessarily ρ(g,r,d,a)≥0. Moreover, any linear series ℓ∈Gr(C) on a curve d of genus g =0,1 satisfies ρ(g,r,d,aℓ(p))≥0 for any point p∈C. For g ≥ 2, pointed curves admitting a linear series with adjusted Brill-Noether number equal to −1 at the marked point form a divisor in M , see [EH89]; when ρ(g,r,d,a) ≤ −2 this locus has g,1 codimension at least 2 in M . In particular, for a general curve C there exists no linear series g,1 ℓ ∈ Gr(C) satisfying aℓ(p) ≥ a for a point p ∈ C if ρ(g,r,d,a) ≤ −2, see [Far13]. It follows that for d eachℓ∈Gr(C),thevanishingsequenceaℓ(p)atanarbitrarypointp∈C satisfiesρ(g,r,d,aℓ(p))≥−1, d and there is at most a finite number of points in C where a linear series ℓ ∈ Gr(C) has vanishing d sequence a verifying ρ(g,r,d,a) = −1. The aim of this note is to determine this number. In the following formula, we let δi be the Kronecker delta and set 1/n!=0, when n<0. j Theorem 1. Fix g ≥ 2 and a : 0 ≤ a < ··· < a ≤ d such that ρ(g,r,d,a) = −1. For a general 0 r curve C of genus g, the number of pairs (p,ℓ)∈C×Gr(C) such that aℓ(p)=a is equal to d a −δj1 −δj2 −a +δj1 +δj2 (4) n :=g! (a −a )2−1 0≤i<k≤r k k k i i i . g,r,d,a 0≤jX1<j2≤r(cid:16) j2 j1 (cid:17)Q ri=0(cid:0)g−d+r+ai−δij1 −δij2 ! (cid:1) Since ρ(g,r,d,a) = −1 and necessarily ρ(g,r,d) Q≥ 0,(cid:0)note that n = 0 in(cid:1) the case a = g,r,d,a (0,1,2,...,r). The caser =1 waspreviouslyknown. Indeed, up to subtracting a basepoint, one can suppose that a = 0. Since ρ(g,1,d,a) = −1, one has d ≥ g +1 and a = 2d−g. In Theorem 1, 0 2 1 we recover the following formula from [HM82, Theorem B] for the number of pencils vanishing with order 2d−g at some unspecified point: g! n =(2d−g−1)(2d−g)(2d−g+1) . g,1,d,(0,2d−g) d!(g−d)! When a = (0,1,...,r−1,r+1) and ρ(g,r,d) = 0, there is only one non-zero summand in the formula for n . We recover the Plu¨cker formula for the total number of ramification points on g,r,d,a every linear series gr on a general curve, see [EH86, pg. 345]: d n =N (r+2)(r+1)r(g−d+r)=N (r+1) d+r(g−1) . g,r,d,a g,r,d g,r,d Letusconsiderthenextnon-trivialexample. Supposeρ(g,r,d)=n(cid:0)−r−1>0,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 ram- ification 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 <···<a ≤d 0 r with ρ(g,r,d,a) = −1. In this section we count the number n of pairs (y,ℓ)∈ C ×Gr(C) such g,r,d,a d that aℓ(y)=a. Note that every such linear series is complete. Let p be a general point of C. Choose m such that the line bundle L⊗O (mp) is non-special for C every L∈Picd(C) (for instance, m=max{2g−2−d+1,0}). The natural evaluation maps H0(L⊗O (mp))→H0(L⊗O (mp)| )։···։H0(L⊗O (mp)| ) C C mp+ary C mp+a0y globalize to π∗(E)→µ (ν∗L⊗O )=:M ։···։µ (ν∗L⊗O )=:M ∗ Dr r ∗ D0 0 as maps of vector bundles over C ×Picd+m(C). Here L is a Poincar´e bundle on C ×Picd+m(C), the map π: C × Picd+m(C) → Picd+m(C) is the second projection, E is a vector bundle of rank d + m − g + 1 defined as E := π (L), the maps µ: C × C × Picd+m(C) → C × Picd+m(C) and ∗ ν: C×C×Picd+m(C)→C×Picd+m(C) are the projections onto the first andthird, andthe second and third factorsrespectively, andfinally O is the structure sheaf ofthe divisorD in C×C whose Di i restriction to {y}×C ∼=C is mp+aiy. Weareinterestedinthelocusofpairs(y,L)suchthath0(L⊗O (−a y))≥r+1−i,fori=0,...,r. C i This is the locus where the morphism of vector bundles ϕ : π∗(E)→M i i has rank at most d+m+i−g−r, for i = 0,...,r. The class of this locus can be computed using Fulton-Pragacz determinantal formula for flag bundles [Ful92, Theorem 10.1]. We shallfirstcompute the Chernpolynomialofthe bundles M . Letπ :C×C×Picd+m(C)→C i i for i = 1,2 and π : C ×C ×Picd+m(C) → Picd+m(C) be the natural projections. Denote by θ 3 the pull-back to C ×C ×Picd+m(C) of the class θ ∈ H2(Picd+m(C)) via π , and denote by η the 3 i cohomologyclassπ∗([point])∈H2(C×C×Picd+m(C)), for i=1,2. Note thatη2 =0. Furthermore, i i given a symplectic basis δ1,...,δ2g for H1(C,Z)∼=H1(Picd+m(C),Z), we denote by δαi the pull-back to C×C×Picd+m(C) of δ via π , for i=1,2,3. Let us define the class α i g γ :=− δjδi −δj δi . i,j s g+s g+s s Xs=1(cid:16) (cid:17) 4 GAVRILFARKASANDNICOLATARASCA Note that γ2 = −2gη η and η γ = γ3 =0, for i=1,2, 1,2 1 2 i 1,2 1,2 γ2 = −2η θ and η γ = γ3 =0, for k=1,2, k,3 k k k,3 k,3 γ γ = η γ , for {i,j}={1,2}. i,j j,3 j i,3 From [ACGH85, §VIII.2], we have ch(ν∗L) = 1+(d+m)η +γ −η θ, 2 2,3 2 ch(O ) = 1−e−(aiη1+aiγ1,2+(ai+m)η2), Di hence via the Grothendieck-Riemann-Rochformula ch(M ) = µ ((1+(1−g)η )·ch(ν∗L⊗O )) i ∗ 2 Di = a +m+η (a2(g−1)+a (d−g+1))+a γ −a η θ. i 1 i i i 1,3 i 1 It follows that the Chern polynomial of M is i c (M )=1+η (a2(g−1)+a (d−g+1))+a γ +(a −a2)η θ. t i 1 i i i 1,3 i i 1 Recall that c (E) =e−tθ ([ACGH85, §VIII.2]). In the following, we will use the Chern classes c(i) := t t c (M −E), that is, t i c(i) =η (a2(g−1)+a (d−g+1))+a γ +θ 1 1 i i i 1,3 and θj a2(g−1)+a (d−g+1) a −a2 a c(i) = +η θj−1 i i + i i + i γ θj−1 j j! 1 (j−1)! (j−2)! (j−1)! 1,3 (cid:18) (cid:19) for j ≥2. From the Fulton-Pragacz formula [Ful92, Theorem 10.1], the number of pairs (y,ℓ) in C ×Gr(C) d with aℓ(y)=a is the degree of the following (r+1)×(r+1) matrix c(r) ··· c(r) g−d+r+ar−r g−d+r+ar c(r−1) c(r−1) ··· c(r−1) (5) n =deg g−d+r+ar−1−r g−d+r+ar−1−(r−1) g−d+r+ar−1 . g,r,d,a  ... ... ...     c(0) ··· c(0)   g−d+r+a0−r g−d+r+a0    Since η2 = η γ = θg+1 = 0, many terms in the expansion of the above determinant are zero. The 1 1 1,3 only terms that survive are the ones obtained by multiplying a summand a2(g−1)+a (d−g+1) a −a2 η θj−1 i i + i i 1 (j−1)! (j−2)! (cid:18) (cid:19) of one of the classes c(i) with r summands θj from the other classes c(i), or the terms obtained by j j! j multiplying two summands a i γ θj−1 1,3 (j−1)! of two different classes c(i) with r−1 summands θj from the other classes c(i). We use the following j j! j variation of the Vandermonde determinant 1 ··· 1 (br−r)! br!  (br−11...−r)! (br−1−1(r−1))! ·.·.·. br−1...1! = Ql<krj(=b0kb−j!bl).  (b0−1r)! ··· b10!  Q   POINTED CASTELNUOVO NUMBERS 5 Hence the quantity (5) can be written as g! (6) n = g,r,d,a r (g−d+r+a )! j=0 j Qr × (a2(g−1)+a (d−g+1))(g−d+r+a ) (a −δi −a +δi)  i i i k k l l i=0 0≤l<k≤r X Y r + (a −a2)(g−d+r+a )(g−d+r+a −1) (a −2δi −a +2δi) i i i i k k l l i=0 0≤l<k≤r X Y −2 a a (g−d+r+a )(g−d+r+a ) i1 i2 i1 i2 0≤iX1<i2≤r (a −δi1 −δi2 −a +δi1 +δi2) k k k l l l  0≤l<k≤r Y  where δi is the Kronecker delta. j Remember that g,r,d,a satisfy the condition ρ(g,r,d,a) = −1. In the following we use the inde- pendent variables r,a ,...,a , and s:=g−d+r. Note that 1 r r r g =rs+s−1+ (a −i), d=rs+r−1+ (a −i). i i i=0 i=0 X X Since the right-hand side of (5) is zero if a =a for any i6=j, we can write (6) as i j (a −a ) (7) n =g! 0≤i<j≤r j i P (r,a)s2+P (r,a)s+P (r,a) g,r,d,a r 2 3 4 (g−d+r+a )! Qj=0 j (cid:16) (cid:17) where Pi(r,a) is a polynomQial in the variables r and a0,...,ar which is symmetric in a0,...,ar for i=2,3,4. Notethattheexpressioninthesquarebracketsin(6)canbereducedtoalinearcombination of the following expressions r at (a −δi −a +δi), i k k l l i=0 l<k X Y r at (a −2δi −a +2δi), i k k l l i=0 l<k X Y (atau+auat) (a −δi −δj −a +δi+δj), i j i j k k k l l l i<j l<k X Y for t,u≥0 such that t+u≤4. From Lemma 1 and Lemma 2 (see below), the polynomial P (r,a) is i symmetric of degree i in a ,...,a and has degree at most i+2 in r, for i=2,3,4. 0 r Since the polynomials P (r,a) are symmetric in a ,...,a , they can be expressed in terms of the i 0 r standard symmetric polynomials in a ,...,a . That is, we can write P (r,a) as a linear combination 0 r i of the finitely many monomials in σ = a , σ = a a , σ = a a a , σ = a a a a 1 i 2 i j 3 i j k 4 i j k l 0≤i≤r 0≤i<j≤r 0≤i<j<k≤r 0≤i<j<k<l≤r X X X X of degree at most i in a ,...,a , with polynomials in r of degree at most i+2 as coefficients. By 0 r the bound on the degree in r, the polynomial P (r,a) is determined by its values at integers r with i 6 GAVRILFARKASANDNICOLATARASCA 1 ≤ r ≤ i+3. Hence, the expression in the square brackets in (6) is determined by its values at integers r with 1≤r ≤7. To complete the proof, it remains to verify the equality of the cumbersome expression for n g,r,d,a in (6) and the compact expression in (4). By pulling out the denominators, the expression in (4) can be rewritten as follows g! (8) r (g−d+r+a )! j=0 j Q × (a −a )2−1 (s+a )(s+a ) (a −δj1 −δj2 −a +δj1 +δj2) .  j2 j1 j1 j2 k k k i i i  0≤jX1<j2≤r(cid:16) (cid:17) 0≤iY<k≤r   Let f be the polynomial in the square brackets in (8), and let h be the polynomial in the s,r,a s,r,a square brackets in (6). By Lemma 2, formula (8) can also be written as in (7), with polynomials P′(r,a) symmetric of degree i in a ,...,a and of degree at most i+2 in r, for i=2,3,4. Hence, to i 0 r show that (8) coincides with (6), it is enough to show that the polynomials f and h coincide s,r,a s,r,a for 1≤r ≤7. When r =1, one has h = (a −a ) (σ2−4σ −1)s2+(σ3−4σ σ −σ )s+σ2σ −4σ2−σ =f . s,1,a 1 0 1 2 1 1 2 1 1 2 2 2 s,1,a (cid:16) (cid:17) Thereafter, one verifies the case r =2: h = (a −a ) (2σ2−6σ −6)s2+(2σ3−7σ σ +9σ +3σ −σ2−4σ +3)s s,2,a j i 1 2 1 1 2 3 2 1 1 0≤Yi<j≤2 (cid:16) +σ2σ −4σ2+3σ σ −σ3−9σ +4σ σ +σ2−5σ +σ −1 1 2 2 1 3 1 3 1 2 1 2 1 = f , (cid:17) s,2,a the case r =3: h = (a −a ) (3σ2−8σ −20)s2+(3σ3−10σ σ +12σ +8σ −3σ2−10σ +20)s s,3,a j i 1 2 1 1 2 3 2 1 1 0≤Yi<j≤3 (cid:16) +σ2σ −4σ2+3σ σ −3σ3−18σ +11σ σ +4σ2−14σ +5σ −10 1 2 2 1 3 1 3 1 2 1 2 1 = f , (cid:17) s,3,a the case r =4: h = (a −a ) (4σ2−10σ −50)s2 s,4,a j i 1 2 0≤Yi<j≤4 (cid:16) +(4σ3−13σ σ +15σ +15σ −6σ2−20σ +75)s 1 1 2 3 2 1 1 +σ2σ −4σ2+3σ σ −6σ3−30σ +21σ σ +10σ2−30σ +15σ −50 1 2 2 1 3 1 3 1 2 1 2 1 = f , (cid:17) s,4,a the case r =5: h = (a −a ) (5σ2−12σ −105)s2 s,5,a j i 1 2 0≤Yi<j≤5 (cid:16) +(5σ3−16σ σ +18σ +24σ −10σ2−35σ +210)s 1 1 2 3 2 1 1 +σ2σ −4σ2+3σ σ −10σ3−45σ +34σ σ +20σ2−55σ +35σ −175 1 2 2 1 3 1 3 1 2 1 2 1 = f , (cid:17) s,5,a POINTED CASTELNUOVO NUMBERS 7 the case r =6: h = (a −a ) (6σ2−14σ −196)s2 s,6,a j i 1 2 0≤Yi<j≤6 (cid:16) +(6σ3−19σ σ +21σ +35σ −15σ2−56σ +490)s 1 1 2 3 2 1 1 +σ2σ −4σ2+3σ σ −15σ3−63σ +50σ σ +35σ2−91σ +70σ −490 1 2 2 1 3 1 3 1 2 1 2 1 = f , (cid:17) s,6,a and, finally, the case r =7: h = (a −a ) (7σ2−16σ −336)s2 s,7,a j i 1 2 0≤Yi<j≤7 (cid:16) +(7σ3−22σ σ +24σ +48σ −21σ2−84σ +1008)s 1 1 2 3 2 1 1 +σ2σ −4σ2+3σ σ −21σ3−84σ +69σ σ +56σ2−140σ +126σ −1176 1 2 2 1 3 1 3 1 2 1 2 1 = f . (cid:17) s,7,a Sinceh =f holdsfor1≤r ≤7,theformulae(6)and(8)coincideforallr. Theorem1follows. s,r,a s,r,a (cid:3) Remark 1. We record the values of the polynomials P (r,a) appearing in the formula (7): i r(r+1)2(r+2) P (r,a) = rσ2−2(r+1)σ − , 2 1 2 12 P (r,a) = rσ3−(3r+1)σ σ +3(r+1)σ 3 1 1 2 3 r(r−1) r(r+1)(r+2) +(r2−1)σ − σ2− σ 2 2 1 6 1 (r−1)r(r+1)2(r+2) + , 24 P (r,a) = σ2σ −4σ2+3σ σ 4 1 2 2 1 3 r(r−1) 3r(r+1) (r−1)(3r+2) − σ3− σ + σ σ 2 1 2 3 2 1 2 (r−1)r(r+1) r(r+1)(2r+1) + σ2− σ 6 1 6 2 (r−1)r(r+1)(r+2) (r−1)r2(r+1)2(r+2) + σ − . 1 24 144 In the above proof, we have used the following two lemmata. Lemma 1. We have r at (a −δi −a +δi)=P(r,a) (a −a ) i k k l l k l i=0 l<k l<k X Y Y where P(r,a) is a polynomial in r and a ,...,a , symmetric of degree t in a ,...,a , and of degree 0 r 0 r at most t+1 in r. Proof. It is easy to see that the left-hand side is anti-symmetric in a ,...,a , hence we can factor by 0 r (a −a ) and obtain a quotient P(r,a) symmetric in a ,...,a . In particular, any monomial in l<k k l 0 r the variables a in the expansion of the left-hand side has degree at least r(r+1). Q i 2 8 GAVRILFARKASANDNICOLATARASCA Let us analyze the expansion of the left-hand side. If we first consider only the summands a −a k l in each factor of each product, we obtain r at (a −a ). i k l ! i=0 l<k X Y This is a homogeneous polynomial in the variables a of degree t + r(r+1) which contributes the i 2 summand r at to P(r,a). i=0 i Next, let us consider non-zero summands of type δi −δi in j factors of each product, and the P l k summands a −a in the remaining factors of each product, for 1≤j ≤r. We obtain k l r (r+1) j (cid:18) (cid:19) homogeneous polynomials in the variables a of degree t+ r(r+1) −j with coefficients all equal to 1. i 2 The sum of such polynomials, if nonzero, is a homogeneous polynomial in the variables a of degree i t+ r(r+1) −j ≥ r(r+1) with coefficients polynomials in r of degree at most j+1. Such polynomial 2 2 contributes a summand to P(r,a) of degree t−j in the variables a and degree at most j+1 in r for i j ≤t, hence the statement. (cid:3) The same result holds for the expressions r at (a −2δi −a +2δi). i k k l l i=0 l<k X Y Example. It is easy to verify the following equality r r r(r+1) a (a −δi −a +δi)= a − (a −a ). i k k l l i 2 k l ! i=0 l<k i=0 l<k X Y X Y Similarly, we have the following. Lemma 2. We have (atau+auat) (a −δi −δj −a +δi+δj)=P(r,a) (a −a ) i j i j k k k l l l k l i<j l<k l<k X Y Y where P(r,a)is a polynomial in r anda ,...,a , symmetricof degreet+u in a ,...,a , and of degree 0 r 0 r at most t+u+2 in r. 2. Classes of pointed Brill-Noether divisors As anapplicationof Theorem1, we compute pointed Brill-Noether divisor classesin M . We fix g,1 a vanishing sequence a : 0 ≤ a < ... < a ≤ d such that ρ(g,r,d,a) = −1 and let Mr (a) be the 0 r g,d locus of smooth curves (C,p) ∈ M admitting a linear series ℓ ∈ Gr(C) having vanishing sequence g,1 d aℓ(p) ≥ a. Eisenbud and Harris proved in [EH89, Theorem 4.1] that the class of the closure of a pointed Brill-Noether divisor Mr (a) in M can be expressed as µBN +νW, where g,d g,1 g−1 g+1 (9) BN :=(g+3)λ− δ − i(g−i)δ irr i 6 i=1 X is the class of the pull-back from M of the Brill-Noether divisor, g g−1 g+1 g−i+1 W :=−λ+ ψ− δ i 2 2 (cid:18) (cid:19) i=1(cid:18) (cid:19) X POINTED CASTELNUOVO NUMBERS 9 is the class of the Weierstrass divisor, and µ and ν are some positive rational numbers. We use the method of test curves to find µ and ν. Let δi be the Kronecker delta. j r Corollary 1. For g >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. 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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: [email protected] University of Utah,Departmentof Mathematics,155S1400E, SaltLake City,UT 84112,USA E-mail address: [email protected]